Classical Invariant Theory#

This module lists classical invariants and covariants of homogeneous polynomials (also called algebraic forms) under the action of the special linear group. That is, we are dealing with polynomials of degree \(d\) in \(n\) variables. The special linear group \(SL(n,\CC)\) acts on the variables \((x_1,\dots, x_n)\) linearly,

\[(x_1,\dots, x_n)^t \to A (x_1,\dots, x_n)^t ,\qquad A \in SL(n,\CC)\]

The linear action on the variables transforms a polynomial \(p\) generally into a different polynomial \(gp\). We can think of it as an action on the space of coefficients in \(p\). An invariant is a polynomial in the coefficients that is invariant under this action. A covariant is a polynomial in the coefficients and the variables \((x_1,\dots, x_n)\) that is invariant under the combined action.

For example, the binary quadratic \(p(x,y) = a x^2 + b x y + c y^2\) has as its invariant the discriminant \(\mathop{disc}(p) = b^2 - 4 a c\). This means that for any \(SL(2,\CC)\) coordinate change

\[\begin{split}\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \qquad \alpha\delta-\beta\gamma=1\end{split}\]

the discriminant is invariant, \(\mathop{disc}\big(p(x',y')\big) = \mathop{disc}\big(p(x,y)\big)\).

To use this module, you should use the factory object invariant_theory. For example, take the quartic:

sage: R.<x,y> = QQ[]
sage: q = x^4 + y^4
sage: quartic = invariant_theory.binary_quartic(q);  quartic
Binary quartic with coefficients (1, 0, 0, 0, 1)

One invariant of a quartic is known as the Eisenstein D-invariant. Since it is an invariant, it is a polynomial in the coefficients (which are integers in this example):

sage: quartic.EisensteinD()
1

One example of a covariant of a quartic is the so-called g-covariant (actually, the Hessian). As with all covariants, it is a polynomial in \(x\), \(y\) and the coefficients:

sage: quartic.g_covariant()
-x^2*y^2

As usual, use tab completion and the online help to discover the implemented invariants and covariants.

In general, the variables of the defining polynomial cannot be guessed. For example, the zero polynomial can be thought of as a homogeneous polynomial of any degree. Also, since we also want to allow polynomial coefficients we cannot just take all variables of the polynomial ring as the variables of the form. This is why you will have to specify the variables explicitly if there is any potential ambiguity. For example:

sage: invariant_theory.binary_quartic(R.zero(), [x,y])
Binary quartic with coefficients (0, 0, 0, 0, 0)

sage: invariant_theory.binary_quartic(x^4, [x,y])
Binary quartic with coefficients (0, 0, 0, 0, 1)

sage: R.<x,y,t> = QQ[]
sage: invariant_theory.binary_quartic(x^4 + y^4 + t*x^2*y^2, [x,y])
Binary quartic with coefficients (1, 0, t, 0, 1)

Finally, it is often convenient to use inhomogeneous polynomials where it is understood that one wants to homogenize them. This is also supported, just define the form with an inhomogeneous polynomial and specify one less variable:

sage: R.<x,t> = QQ[]
sage: invariant_theory.binary_quartic(x^4 + 1 + t*x^2, [x])
Binary quartic with coefficients (1, 0, t, 0, 1)

REFERENCES:

Wikipedia article Glossary_of_invariant_theory

AUTHORS:

Volker Braun (2013-01-24): initial version
Jesper Noordsij (2018-05-18): support for binary quintics added

class sage.rings.invariants.invariant_theory.AlgebraicForm(n, d, polynomial, *args, **kwds)#

Bases: FormsBase

The base class of algebraic forms (i.e. homogeneous polynomials).

You should only instantiate the derived classes of this base class.

Derived classes must implement coeffs() and scaled_coeffs()

INPUT:

n – The number of variables.
d – The degree of the polynomial.
polynomial – The polynomial.
*args – The variables, as a single list/tuple, multiple arguments, or None to use all variables of the polynomial.

Derived classes must implement the same arguments for the constructor.

EXAMPLES:

sage: from sage.rings.invariants.invariant_theory import AlgebraicForm
sage: R.<x,y> = QQ[]
sage: p = x^2 + y^2
sage: AlgebraicForm(2, 2, p).variables()
(x, y)
sage: AlgebraicForm(2, 2, p, None).variables()
(x, y)
sage: AlgebraicForm(3, 2, p).variables()
(x, y, None)
sage: AlgebraicForm(3, 2, p, None).variables()
(x, y, None)

sage: from sage.rings.invariants.invariant_theory import AlgebraicForm
sage: R.<x,y,s,t> = QQ[]
sage: p = s*x^2 + t*y^2
sage: AlgebraicForm(2, 2, p, [x,y]).variables()
(x, y)
sage: AlgebraicForm(2, 2, p, x,y).variables()
(x, y)

sage: AlgebraicForm(3, 2, p, [x,y,None]).variables()
(x, y, None)
sage: AlgebraicForm(3, 2, p, x,y,None).variables()
(x, y, None)

sage: AlgebraicForm(2, 1, p, [x,y]).variables()
Traceback (most recent call last):
...
ValueError: polynomial is of the wrong degree

sage: AlgebraicForm(2, 2, x^2+y, [x,y]).variables()
Traceback (most recent call last):
...
ValueError: polynomial is not homogeneous

coefficients()#

Alias for coeffs().

See the documentation for coeffs() for details.

EXAMPLES:

sage: R.<a,b,c,d,e,f,g, x,y,z> = QQ[]
sage: p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z
sage: q = invariant_theory.quadratic_form(p, x,y,z)
sage: q.coefficients()
(a, b, c, d, e, f)
sage: q.coeffs()
(a, b, c, d, e, f)

form()#

Return the defining polynomial.

OUTPUT:

The polynomial used to define the algebraic form.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+y^4)
sage: quartic.form()
x^4 + y^4
sage: quartic.polynomial()
x^4 + y^4

homogenized(var='h')#

Return form as defined by a homogeneous polynomial.

INPUT:

var – either a variable name, variable index or a variable (default: 'h').

OUTPUT:

The same algebraic form, but defined by a homogeneous polynomial.

EXAMPLES:

sage: T.<t> = QQ[]
sage: quadratic = invariant_theory.binary_quadratic(t^2 + 2*t + 3)
sage: quadratic
Binary quadratic with coefficients (1, 3, 2)
sage: quadratic.homogenized()
Binary quadratic with coefficients (1, 3, 2)
sage: quadratic == quadratic.homogenized()
True
sage: quadratic.form()
t^2 + 2*t + 3
sage: quadratic.homogenized().form()
t^2 + 2*t*h + 3*h^2

sage: R.<x,y,z> = QQ[]
sage: quadratic = invariant_theory.ternary_quadratic(x^2 + 1, [x,y])
sage: quadratic.homogenized().form()
x^2 + h^2

sage: R.<x> = QQ[]
sage: quintic = invariant_theory.binary_quintic(x^4 + 1, x)
sage: quintic.homogenized().form()
x^4*h + h^5

polynomial()#

Return the defining polynomial.

OUTPUT:

The polynomial used to define the algebraic form.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+y^4)
sage: quartic.form()
x^4 + y^4
sage: quartic.polynomial()
x^4 + y^4

transformed(g)#

Return the image under a linear transformation of the variables.

INPUT:

g – a \(GL(n,\CC)\) matrix or a dictionary with the
variables as keys. A matrix is used to define the linear transformation of homogeneous variables, a dictionary acts by substitution of the variables.

OUTPUT:

A new instance of a subclass of AlgebraicForm obtained by replacing the variables of the homogeneous polynomial by their image under g.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3 + 2*y^3 + 3*z^3 + 4*x*y*z)
sage: cubic.transformed({x:y, y:z, z:x}).form()
3*x^3 + y^3 + 4*x*y*z + 2*z^3
sage: cyc = matrix([[0,1,0],[0,0,1],[1,0,0]])
sage: cubic.transformed(cyc) == cubic.transformed({x:y, y:z, z:x})
True
sage: g = matrix(QQ, [[1, 0, 0], [-1, 1, -3], [-5, -5, 16]])
sage: cubic.transformed(g)
Ternary cubic with coefficients (-356, -373, 12234, -1119, 3578, -1151,
3582, -11766, -11466, 7360)
sage: cubic.transformed(g).transformed(g.inverse()) == cubic
True

class sage.rings.invariants.invariant_theory.BinaryQuartic(n, d, polynomial, *args)#

Bases: AlgebraicForm

Invariant theory of a binary quartic.

You should use the invariant_theory factory object to construct instances of this class. See binary_quartic() for details.

EisensteinD()#

One of the Eisenstein invariants of a binary quartic.

OUTPUT:

The Eisenstein D-invariant of the quartic.

\[\begin{split}f(x) = a_0 x_1^4 + 4 a_1 x_0 x_1^3 + 6 a_2 x_0^2 x_1^2 + 4 a_3 x_0^3 x_1 + a_4 x_0^4 \\ \Rightarrow D(f) = a_0 a_4+3 a_2^2-4 a_1 a_3\end{split}\]

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, x0, x1> = QQ[]
sage: f = a0*x1^4+4*a1*x0*x1^3+6*a2*x0^2*x1^2+4*a3*x0^3*x1+a4*x0^4
sage: inv = invariant_theory.binary_quartic(f, x0, x1)
sage: inv.EisensteinD()
3*a2^2 - 4*a1*a3 + a0*a4

EisensteinE()#

One of the Eisenstein invariants of a binary quartic.

OUTPUT:

The Eisenstein E-invariant of the quartic.

\[\begin{split}f(x) = a_0 x_1^4 + 4 a_1 x_0 x_1^3 + 6 a_2 x_0^2 x_1^2 + 4 a_3 x_0^3 x_1 + a_4 x_0^4 \\ \Rightarrow E(f) = a_0 a_3^2 +a_1^2 a_4 -a_0 a_2 a_4 -2 a_1 a_2 a_3 + a_2^3\end{split}\]

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, x0, x1> = QQ[]
sage: f = a0*x1^4+4*a1*x0*x1^3+6*a2*x0^2*x1^2+4*a3*x0^3*x1+a4*x0^4
sage: inv = invariant_theory.binary_quartic(f, x0, x1)
sage: inv.EisensteinE()
a2^3 - 2*a1*a2*a3 + a0*a3^2 + a1^2*a4 - a0*a2*a4

coeffs()#

The coefficients of a binary quartic.

Given

\[f(x) = a_0 x_1^4 + a_1 x_0 x_1^3 + a_2 x_0^2 x_1^2 + a_3 x_0^3 x_1 + a_4 x_0^4\]

this function returns \(a = (a_0, a_1, a_2, a_3, a_4)\)

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, x0, x1> = QQ[]
sage: p = a0*x1^4 + a1*x1^3*x0 + a2*x1^2*x0^2 + a3*x1*x0^3 + a4*x0^4
sage: quartic = invariant_theory.binary_quartic(p, x0, x1)
sage: quartic.coeffs()
(a0, a1, a2, a3, a4)

sage: R.<a0, a1, a2, a3, a4, x> = QQ[]
sage: p = a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4
sage: quartic = invariant_theory.binary_quartic(p, x)
sage: quartic.coeffs()
(a0, a1, a2, a3, a4)

g_covariant()#

The g-covariant of a binary quartic.

OUTPUT:

The g-covariant of the quartic.

\[\begin{split}f(x) = a_0 x_1^4 + 4 a_1 x_0 x_1^3 + 6 a_2 x_0^2 x_1^2 + 4 a_3 x_0^3 x_1 + a_4 x_0^4 \\ \Rightarrow D(f) = \frac{1}{144} \begin{pmatrix} \frac{\partial^2 f}{\partial x \partial x} \end{pmatrix}\end{split}\]

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, x, y> = QQ[]
sage: p = a0*x^4+4*a1*x^3*y+6*a2*x^2*y^2+4*a3*x*y^3+a4*y^4
sage: inv = invariant_theory.binary_quartic(p, x, y)
sage: g = inv.g_covariant();  g
a1^2*x^4 - a0*a2*x^4 + 2*a1*a2*x^3*y - 2*a0*a3*x^3*y + 3*a2^2*x^2*y^2
- 2*a1*a3*x^2*y^2 - a0*a4*x^2*y^2 + 2*a2*a3*x*y^3
- 2*a1*a4*x*y^3 + a3^2*y^4 - a2*a4*y^4

sage: inv_inhomogeneous = invariant_theory.binary_quartic(p.subs(y=1), x)
sage: inv_inhomogeneous.g_covariant()
a1^2*x^4 - a0*a2*x^4 + 2*a1*a2*x^3 - 2*a0*a3*x^3 + 3*a2^2*x^2
- 2*a1*a3*x^2 - a0*a4*x^2 + 2*a2*a3*x - 2*a1*a4*x + a3^2 - a2*a4

sage: g == 1/144 * (p.derivative(x,y)^2 - p.derivative(x,x)*p.derivative(y,y))
True

h_covariant()#

The h-covariant of a binary quartic.

OUTPUT:

The h-covariant of the quartic.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, x, y> = QQ[]
sage: p = a0*x^4+4*a1*x^3*y+6*a2*x^2*y^2+4*a3*x*y^3+a4*y^4
sage: inv = invariant_theory.binary_quartic(p, x, y)
sage: h = inv.h_covariant();   h
-2*a1^3*x^6 + 3*a0*a1*a2*x^6 - a0^2*a3*x^6 - 6*a1^2*a2*x^5*y + 9*a0*a2^2*x^5*y
- 2*a0*a1*a3*x^5*y - a0^2*a4*x^5*y - 10*a1^2*a3*x^4*y^2 + 15*a0*a2*a3*x^4*y^2
- 5*a0*a1*a4*x^4*y^2 + 10*a0*a3^2*x^3*y^3 - 10*a1^2*a4*x^3*y^3
+ 10*a1*a3^2*x^2*y^4 - 15*a1*a2*a4*x^2*y^4 + 5*a0*a3*a4*x^2*y^4
+ 6*a2*a3^2*x*y^5 - 9*a2^2*a4*x*y^5 + 2*a1*a3*a4*x*y^5 + a0*a4^2*x*y^5
+ 2*a3^3*y^6 - 3*a2*a3*a4*y^6 + a1*a4^2*y^6

sage: inv_inhomogeneous = invariant_theory.binary_quartic(p.subs(y=1), x)
sage: inv_inhomogeneous.h_covariant()
-2*a1^3*x^6 + 3*a0*a1*a2*x^6 - a0^2*a3*x^6 - 6*a1^2*a2*x^5 + 9*a0*a2^2*x^5
- 2*a0*a1*a3*x^5 - a0^2*a4*x^5 - 10*a1^2*a3*x^4 + 15*a0*a2*a3*x^4
- 5*a0*a1*a4*x^4 + 10*a0*a3^2*x^3 - 10*a1^2*a4*x^3 + 10*a1*a3^2*x^2
- 15*a1*a2*a4*x^2 + 5*a0*a3*a4*x^2 + 6*a2*a3^2*x - 9*a2^2*a4*x
+ 2*a1*a3*a4*x + a0*a4^2*x + 2*a3^3 - 3*a2*a3*a4 + a1*a4^2

sage: g = inv.g_covariant()
sage: h == 1/8 * (p.derivative(x)*g.derivative(y)-p.derivative(y)*g.derivative(x))
True

monomials()#

List the basis monomials in the form.

OUTPUT:

A tuple of monomials. They are in the same order as coeffs().

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+y^4)
sage: quartic.monomials()
(y^4, x*y^3, x^2*y^2, x^3*y, x^4)

scaled_coeffs()#

The coefficients of a binary quartic.

Given

\[f(x) = a_0 x_1^4 + 4 a_1 x_0 x_1^3 + 6 a_2 x_0^2 x_1^2 + 4 a_3 x_0^3 x_1 + a_4 x_0^4\]

this function returns \(a = (a_0, a_1, a_2, a_3, a_4)\)

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, x0, x1> = QQ[]
sage: quartic = a0*x1^4 + 4*a1*x1^3*x0 + 6*a2*x1^2*x0^2 + 4*a3*x1*x0^3 + a4*x0^4
sage: inv = invariant_theory.binary_quartic(quartic, x0, x1)
sage: inv.scaled_coeffs()
(a0, a1, a2, a3, a4)

sage: R.<a0, a1, a2, a3, a4, x> = QQ[]
sage: quartic = a0 + 4*a1*x + 6*a2*x^2 + 4*a3*x^3 + a4*x^4
sage: inv = invariant_theory.binary_quartic(quartic, x)
sage: inv.scaled_coeffs()
(a0, a1, a2, a3, a4)

class sage.rings.invariants.invariant_theory.BinaryQuintic(n, d, polynomial, *args)#

Bases: AlgebraicForm

Invariant theory of a binary quintic form.

You should use the invariant_theory factory object to construct instances of this class. See binary_quintic() for details.

REFERENCES:

For a description of all invariants and covariants of a binary quintic, see section 73 of [Cle1872].

A_invariant()#

Return the invariant \(A\) of a binary quintic.

OUTPUT:

The \(A\)-invariant of the binary quintic.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.A_invariant()
4/625*a2^2*a3^2 - 12/625*a1*a3^3 - 12/625*a2^3*a4
+ 38/625*a1*a2*a3*a4 + 6/125*a0*a3^2*a4 - 18/625*a1^2*a4^2
- 16/125*a0*a2*a4^2 + 6/125*a1*a2^2*a5 - 16/125*a1^2*a3*a5
- 2/25*a0*a2*a3*a5 + 4/5*a0*a1*a4*a5 - 2*a0^2*a5^2

B_invariant()#

Return the invariant \(B\) of a binary quintic.

OUTPUT:

The \(B\)-invariant of the binary quintic.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.B_invariant()
1/1562500*a2^4*a3^4 - 3/781250*a1*a2^2*a3^5 + 9/1562500*a1^2*a3^6
- 3/781250*a2^5*a3^2*a4 + 37/1562500*a1*a2^3*a3^3*a4
- 57/1562500*a1^2*a2*a3^4*a4 + 3/312500*a0*a2^2*a3^4*a4
...
+ 8/625*a0^2*a1^2*a4^2*a5^2 - 4/125*a0^3*a2*a4^2*a5^2 - 16/3125*a1^5*a5^3
+ 4/125*a0*a1^3*a2*a5^3 - 6/125*a0^2*a1*a2^2*a5^3
- 4/125*a0^2*a1^2*a3*a5^3 + 2/25*a0^3*a2*a3*a5^3

C_invariant()#

Return the invariant \(C\) of a binary quintic.

OUTPUT:

The \(C\)-invariant of the binary quintic.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.C_invariant()
-3/1953125000*a2^6*a3^6 + 27/1953125000*a1*a2^4*a3^7
- 249/7812500000*a1^2*a2^2*a3^8 - 3/78125000*a0*a2^3*a3^8
+ 3/976562500*a1^3*a3^9 + 27/156250000*a0*a1*a2*a3^9
...
+ 192/15625*a0^2*a1^3*a2^2*a3*a5^4 - 36/3125*a0^3*a1*a2^3*a3*a5^4
+ 24/15625*a0^2*a1^4*a3^2*a5^4 - 24/3125*a0^3*a1^2*a2*a3^2*a5^4
+ 6/625*a0^4*a2^2*a3^2*a5^4

H_covariant(as_form=False)#

Return the covariant \(H\) of a binary quintic.

INPUT:

as_form – if as_form is False, the result will be returned as polynomial (default). If it is True the result is returned as an object of the class AlgebraicForm.

OUTPUT:

The \(H\)-covariant of the binary quintic as polynomial or as binary form.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.H_covariant()
-2/25*a4^2*x0^6 + 1/5*a3*a5*x0^6 - 3/25*a3*a4*x0^5*x1
+ 3/5*a2*a5*x0^5*x1 - 3/25*a3^2*x0^4*x1^2 + 3/25*a2*a4*x0^4*x1^2
+ 6/5*a1*a5*x0^4*x1^2 - 4/25*a2*a3*x0^3*x1^3 + 14/25*a1*a4*x0^3*x1^3
+ 2*a0*a5*x0^3*x1^3 - 3/25*a2^2*x0^2*x1^4 + 3/25*a1*a3*x0^2*x1^4
+ 6/5*a0*a4*x0^2*x1^4 - 3/25*a1*a2*x0*x1^5 + 3/5*a0*a3*x0*x1^5
- 2/25*a1^2*x1^6 + 1/5*a0*a2*x1^6

sage: quintic.H_covariant(as_form=True)
Binary sextic given by ...

R_invariant()#

Return the invariant \(R\) of a binary quintic.

OUTPUT:

The \(R\)-invariant of the binary quintic.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.R_invariant()
3/3906250000000*a1^2*a2^5*a3^11 - 3/976562500000*a0*a2^6*a3^11
- 51/7812500000000*a1^3*a2^3*a3^12 + 27/976562500000*a0*a1*a2^4*a3^12
+ 27/1953125000000*a1^4*a2*a3^13 - 81/1562500000000*a0*a1^2*a2^2*a3^13
...
+ 384/9765625*a0*a1^10*a5^7 - 192/390625*a0^2*a1^8*a2*a5^7
+ 192/78125*a0^3*a1^6*a2^2*a5^7 - 96/15625*a0^4*a1^4*a2^3*a5^7
+ 24/3125*a0^5*a1^2*a2^4*a5^7 - 12/3125*a0^6*a2^5*a5^7

T_covariant(as_form=False)#

Return the covariant \(T\) of a binary quintic.

INPUT:

as_form – if as_form is False, the result will be returned as polynomial (default). If it is True the result is returned as an object of the class AlgebraicForm.

OUTPUT:

The \(T\)-covariant of the binary quintic as polynomial or as binary form.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.T_covariant()
2/125*a4^3*x0^9 - 3/50*a3*a4*a5*x0^9 + 1/10*a2*a5^2*x0^9
+ 9/250*a3*a4^2*x0^8*x1 - 3/25*a3^2*a5*x0^8*x1 + 1/50*a2*a4*a5*x0^8*x1
+ 2/5*a1*a5^2*x0^8*x1 + 3/250*a3^2*a4*x0^7*x1^2 + 8/125*a2*a4^2*x0^7*x1^2
...
11/25*a0*a1*a4*x0^2*x1^7 - a0^2*a5*x0^2*x1^7 - 9/250*a1^2*a2*x0*x1^8
+ 3/25*a0*a2^2*x0*x1^8 - 1/50*a0*a1*a3*x0*x1^8 - 2/5*a0^2*a4*x0*x1^8
- 2/125*a1^3*x1^9 + 3/50*a0*a1*a2*x1^9 - 1/10*a0^2*a3*x1^9

sage: quintic.T_covariant(as_form=True)
Binary nonic given by ...

alpha_covariant(as_form=False)#

Return the covariant \(\alpha\) of a binary quintic.

INPUT:

as_form – if as_form is False, the result will be returned as polynomial (default). If it is True the result is returned as an object of the class AlgebraicForm.

OUTPUT:

The \(\alpha\)-covariant of the binary quintic as polynomial or as binary form.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.alpha_covariant()
1/2500*a2^2*a3^3*x0 - 3/2500*a1*a3^4*x0 - 1/625*a2^3*a3*a4*x0
+ 3/625*a1*a2*a3^2*a4*x0 + 3/625*a0*a3^3*a4*x0 + 2/625*a1*a2^2*a4^2*x0
- 6/625*a1^2*a3*a4^2*x0 - 12/625*a0*a2*a3*a4^2*x0 + 24/625*a0*a1*a4^3*x0
...
- 12/625*a1^2*a2*a3*a5*x1 - 1/125*a0*a2^2*a3*a5*x1
+ 8/125*a0*a1*a3^2*a5*x1 + 24/625*a1^3*a4*a5*x1 - 8/125*a0*a1*a2*a4*a5*x1
- 4/25*a0^2*a3*a4*a5*x1 - 4/25*a0*a1^2*a5^2*x1 + 2/5*a0^2*a2*a5^2*x1

sage: quintic.alpha_covariant(as_form=True)
Binary monic given by ...

arithmetic_invariants()#

Return a set of generating arithmetic invariants of a binary quintic.

An arithmetic invariants is an invariant whose coefficients are integers for a general binary quintic. They are linear combinations of the Clebsch invariants, such that they still generate the ring of invariants.

OUTPUT:

The arithmetic invariants of the binary quintic. They are given by

\[\begin{split}\begin{aligned} I_4 & = 2^{-1} \cdot 5^4 \cdot A \\ I_8 & = 5^5 \cdot (2^{-1} \cdot 47 \cdot A^2 - 2^2 \cdot B) \\ I_{12} & = 5^{10} \cdot (2^{-1} \cdot 3 \cdot A^3 - 2^5 \cdot 3^{-1} \cdot C) \\ I_{18} & = 2^8 \cdot 3^{-1} \cdot 5^{15} \cdot R \\ \end{aligned}\end{split}\]

where \(A\), \(B\), \(C\) and \(R\) are the BinaryQuintic.clebsch_invariants().

EXAMPLES:

sage: R.<x0, x1> = QQ[]
sage: p = 2*x1^5 + 4*x1^4*x0 + 5*x1^3*x0^2 + 7*x1^2*x0^3 - 11*x1*x0^4 + x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.arithmetic_invariants()
{'I12': -1156502613073152,
 'I18': -12712872348048797642752,
 'I4': -138016,
 'I8': 14164936192}

We can check that the coefficients of the invariants have no common divisor for a general quintic form:

sage: R.<a0,a1,a2,a3,a4,a5,x0,x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: invs = quintic.arithmetic_invariants()
sage: [invs[x].content() for x in invs]
[1, 1, 1, 1]

beta_covariant(as_form=False)#

Return the covariant \(\beta\) of a binary quintic.

INPUT:

as_form – if as_form is False, the result will be returned as polynomial (default). If it is True the result is returned as an object of the class AlgebraicForm.

OUTPUT:

The \(\beta\)-covariant of the binary quintic as polynomial or as binary form.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.beta_covariant()
-1/62500*a2^3*a3^4*x0 + 9/125000*a1*a2*a3^5*x0 - 27/125000*a0*a3^6*x0
+ 13/125000*a2^4*a3^2*a4*x0 - 31/62500*a1*a2^2*a3^3*a4*x0
- 3/62500*a1^2*a3^4*a4*x0 + 27/15625*a0*a2*a3^4*a4*x0
...
- 16/125*a0^2*a1*a3^2*a5^2*x1 - 28/625*a0*a1^3*a4*a5^2*x1
+ 6/125*a0^2*a1*a2*a4*a5^2*x1 + 8/25*a0^3*a3*a4*a5^2*x1
+ 4/25*a0^2*a1^2*a5^3*x1 - 2/5*a0^3*a2*a5^3*x1

sage: quintic.beta_covariant(as_form=True)
Binary monic given by ...

canonical_form(reduce_gcd=False)#

Return a canonical representative of the quintic.

Given a binary quintic \(f\) with coefficients in a field \(K\), returns a canonical representative of the \(GL(2,\bar{K})\)-orbit of the quintic, where \(\bar{K}\) is an algebraic closure of \(K\). This means that two binary quintics \(f\) and \(g\) are \(GL(2,\bar{K})\)-equivalent if and only if their canonical forms are the same.

INPUT:

reduce_gcd – If set to True, then a variant of this canonical form is computed where the coefficients are coprime integers. The obtained form is then unique up to multiplication by a unit. See also binary_quintic_from_invariants()’.

OUTPUT:

A canonical \(GL(2,\bar{K})\)-equivalent binary quintic.

EXAMPLES:

sage: R.<x0, x1> = QQ[]
sage: p = 2*x1^5 + 4*x1^4*x0 + 5*x1^3*x0^2 + 7*x1^2*x0^3 - 11*x1*x0^4 + x0^5
sage: f = invariant_theory.binary_quintic(p, x0, x1)
sage: g = matrix(QQ, [[11,5],[7,2]])
sage: gf = f.transformed(g)
sage: f.canonical_form() == gf.canonical_form()
True
sage: h = f.canonical_form(reduce_gcd=True)
sage: gcd(h.coeffs())
1

clebsch_invariants(as_tuple=False)#

Return the invariants of a binary quintic as described by Clebsch.

The following invariants are returned: \(A\), \(B\), \(C\) and \(R\).

OUTPUT:

The Clebsch invariants of the binary quintic.

EXAMPLES:

sage: R.<x0, x1> = QQ[]
sage: p = 2*x1^5 + 4*x1^4*x0 + 5*x1^3*x0^2 + 7*x1^2*x0^3 - 11*x1*x0^4 + x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.clebsch_invariants()
{'A': -276032/625,
 'B': 4983526016/390625,
 'C': -247056495846408/244140625,
 'R': -148978972828696847376/30517578125}

sage: quintic.clebsch_invariants(as_tuple=True)
(-276032/625,
 4983526016/390625,
 -247056495846408/244140625,
 -148978972828696847376/30517578125)

coeffs()#

The coefficients of a binary quintic.

Given

\[f(x) = a_0 x_1^5 + a_1 x_0 x_1^4 + a_2 x_0^2 x_1^3 + a_3 x_0^3 x_1^2 + a_4 x_0^4 x_1 + a_5 x_1^5\]

this function returns \(a = (a_0, a_1, a_2, a_3, a_4, a_5)\)

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.coeffs()
(a0, a1, a2, a3, a4, a5)

sage: R.<a0, a1, a2, a3, a4, a5, x> = QQ[]
sage: p = a0 + a1*x + a2*x^2 + a3*x^3 + a4*x^4 + a5*x^5
sage: quintic = invariant_theory.binary_quintic(p, x)
sage: quintic.coeffs()
(a0, a1, a2, a3, a4, a5)

delta_covariant(as_form=False)#

Return the covariant \(\delta\) of a binary quintic.

INPUT:

as_form – if as_form is False, the result will be returned as polynomial (default). If it is True the result is returned as an object of the class AlgebraicForm.

OUTPUT:

The \(\delta\)-covariant of the binary quintic as polynomial or as binary form.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.delta_covariant()
1/1562500000*a2^6*a3^7*x0 - 9/1562500000*a1*a2^4*a3^8*x0
+ 9/625000000*a1^2*a2^2*a3^9*x0 + 9/781250000*a0*a2^3*a3^9*x0
- 9/1562500000*a1^3*a3^10*x0 - 81/1562500000*a0*a1*a2*a3^10*x0
...
+ 64/3125*a0^3*a1^3*a2^2*a5^5*x1 - 12/625*a0^4*a1*a2^3*a5^5*x1
+ 16/3125*a0^3*a1^4*a3*a5^5*x1 - 16/625*a0^4*a1^2*a2*a3*a5^5*x1
+ 4/125*a0^5*a2^2*a3*a5^5*x1

sage: quintic.delta_covariant(as_form=True)
Binary monic given by ...

classmethod from_invariants(invariants, x, z, *args, **kwargs)#

Construct a binary quintic from its invariants.

This function constructs a binary quintic whose invariants equal the ones provided as argument up to scaling.

INPUT:

invariants – A list or tuple of invariants that are used to reconstruct the binary quintic.

OUTPUT:

A BinaryQuintic.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: from sage.rings.invariants.invariant_theory import BinaryQuintic
sage: BinaryQuintic.from_invariants([3,6,12], x, y)
Binary quintic with coefficients (0, 1, 0, 0, 1, 0)

gamma_covariant(as_form=False)#

Return the covariant \(\gamma\) of a binary quintic.

INPUT:

as_form – if as_form is False, the result will be returned as polynomial (default). If it is True the result is returned as an object of the class AlgebraicForm.

OUTPUT:

The \(\gamma\)-covariant of the binary quintic as polynomial or as binary form.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.gamma_covariant()
1/156250000*a2^5*a3^6*x0 - 3/62500000*a1*a2^3*a3^7*x0
+ 27/312500000*a1^2*a2*a3^8*x0 + 27/312500000*a0*a2^2*a3^8*x0
- 81/312500000*a0*a1*a3^9*x0 - 19/312500000*a2^6*a3^4*a4*x0
...
- 32/3125*a0^2*a1^3*a2^2*a5^4*x1 + 6/625*a0^3*a1*a2^3*a5^4*x1
- 8/3125*a0^2*a1^4*a3*a5^4*x1 + 8/625*a0^3*a1^2*a2*a3*a5^4*x1
- 2/125*a0^4*a2^2*a3*a5^4*x1

sage: quintic.gamma_covariant(as_form=True)
Binary monic given by ...

i_covariant(as_form=False)#

Return the covariant \(i\) of a binary quintic.

INPUT:

as_form – if as_form is False, the result will be returned as polynomial (default). If it is True the result is returned as an object of the class AlgebraicForm.

OUTPUT:

The \(i\)-covariant of the binary quintic as polynomial or as binary form.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.i_covariant()
3/50*a3^2*x0^2 - 4/25*a2*a4*x0^2 + 2/5*a1*a5*x0^2 + 1/25*a2*a3*x0*x1
- 6/25*a1*a4*x0*x1 + 2*a0*a5*x0*x1 + 3/50*a2^2*x1^2 - 4/25*a1*a3*x1^2
+ 2/5*a0*a4*x1^2

sage: quintic.i_covariant(as_form=True)
Binary quadratic given by ...

invariants(type='clebsch')#

Return a tuple of invariants of a binary quintic.

INPUT:

type – The type of invariants to return. The default choice is to return the Clebsch invariants.

OUTPUT:

The invariants of the binary quintic.

EXAMPLES:

sage: R.<x0, x1> = QQ[]
sage: p = 2*x1^5 + 4*x1^4*x0 + 5*x1^3*x0^2 + 7*x1^2*x0^3 - 11*x1*x0^4 + x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.invariants()
(-276032/625,
 4983526016/390625,
 -247056495846408/244140625,
 -148978972828696847376/30517578125)
sage: quintic.invariants('unknown')
Traceback (most recent call last):
...
ValueError: unknown type of invariants unknown for a binary quintic

j_covariant(as_form=False)#

Return the covariant \(j\) of a binary quintic.

INPUT:

as_form – if as_form is False, the result will be returned as polynomial (default). If it is True the result is returned as an object of the class AlgebraicForm.

OUTPUT:

The \(j\)-covariant of the binary quintic as polynomial or as binary form.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.j_covariant()
-3/500*a3^3*x0^3 + 3/125*a2*a3*a4*x0^3 - 6/125*a1*a4^2*x0^3
- 3/50*a2^2*a5*x0^3 + 3/25*a1*a3*a5*x0^3 - 3/500*a2*a3^2*x0^2*x1
+ 3/250*a2^2*a4*x0^2*x1 + 3/125*a1*a3*a4*x0^2*x1 - 6/25*a0*a4^2*x0^2*x1
- 3/25*a1*a2*a5*x0^2*x1 + 3/5*a0*a3*a5*x0^2*x1 - 3/500*a2^2*a3*x0*x1^2
+ 3/250*a1*a3^2*x0*x1^2 + 3/125*a1*a2*a4*x0*x1^2 - 3/25*a0*a3*a4*x0*x1^2
- 6/25*a1^2*a5*x0*x1^2 + 3/5*a0*a2*a5*x0*x1^2 - 3/500*a2^3*x1^3
+ 3/125*a1*a2*a3*x1^3 - 3/50*a0*a3^2*x1^3 - 6/125*a1^2*a4*x1^3
+ 3/25*a0*a2*a4*x1^3

sage: quintic.j_covariant(as_form=True)
Binary cubic given by ...

monomials()#

List the basis monomials of the form.

This function lists a basis of monomials of the space of binary quintics of which this form is an element.

OUTPUT:

A tuple of monomials. They are in the same order as coeffs().

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: quintic = invariant_theory.binary_quintic(x^5+y^5)
sage: quintic.monomials()
(y^5, x*y^4, x^2*y^3, x^3*y^2, x^4*y, x^5)

scaled_coeffs()#

The coefficients of a binary quintic.

Given

\[f(x) = a_0 x_1^5 + 5 a_1 x_0 x_1^4 + 10 a_2 x_0^2 x_1^3 + 10 a_3 x_0^3 x_1^2 + 5 a_4 x_0^4 x_1 + a_5 x_1^5\]

this function returns \(a = (a_0, a_1, a_2, a_3, a_4, a_5)\)

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + 5*a1*x1^4*x0 + 10*a2*x1^3*x0^2 + 10*a3*x1^2*x0^3 + 5*a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.scaled_coeffs()
(a0, a1, a2, a3, a4, a5)

sage: R.<a0, a1, a2, a3, a4, a5, x> = QQ[]
sage: p = a0 + 5*a1*x + 10*a2*x^2 + 10*a3*x^3 + 5*a4*x^4 + a5*x^5
sage: quintic = invariant_theory.binary_quintic(p, x)
sage: quintic.scaled_coeffs()
(a0, a1, a2, a3, a4, a5)

tau_covariant(as_form=False)#

Return the covariant \(\tau\) of a binary quintic.

INPUT:

as_form – if as_form is False, the result will be returned as polynomial (default). If it is True the result is returned as an object of the class AlgebraicForm.

OUTPUT:

The \(\tau\)-covariant of the binary quintic as polynomial or as binary form.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.tau_covariant()
1/62500*a2^2*a3^4*x0^2 - 3/62500*a1*a3^5*x0^2
- 1/15625*a2^3*a3^2*a4*x0^2 + 1/6250*a1*a2*a3^3*a4*x0^2
+ 3/6250*a0*a3^4*a4*x0^2 - 1/31250*a2^4*a4^2*x0^2
...
- 2/125*a0*a1*a2^2*a4*a5*x1^2 - 4/125*a0*a1^2*a3*a4*a5*x1^2
+ 2/25*a0^2*a2*a3*a4*a5*x1^2 - 8/625*a1^4*a5^2*x1^2
+ 8/125*a0*a1^2*a2*a5^2*x1^2 - 2/25*a0^2*a2^2*a5^2*x1^2

sage: quintic.tau_covariant(as_form=True)
Binary quadratic given by ...

theta_covariant(as_form=False)#

Return the covariant \(\theta\) of a binary quintic.

INPUT:

as_form – if as_form is False, the result will be returned as polynomial (default). If it is True the result is returned as an object of the class AlgebraicForm.

OUTPUT:

The \(\theta\)-covariant of the binary quintic as polynomial or as binary form.

EXAMPLES:

sage: R.<a0, a1, a2, a3, a4, a5, x0, x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: quintic.theta_covariant()
-1/625000*a2^3*a3^5*x0^2 + 9/1250000*a1*a2*a3^6*x0^2
- 27/1250000*a0*a3^7*x0^2 + 3/250000*a2^4*a3^3*a4*x0^2
- 7/125000*a1*a2^2*a3^4*a4*x0^2 - 3/312500*a1^2*a3^5*a4*x0^2
...
+ 6/625*a0^2*a1*a2^2*a4*a5^2*x1^2 + 24/625*a0^2*a1^2*a3*a4*a5^2*x1^2
- 12/125*a0^3*a2*a3*a4*a5^2*x1^2 + 8/625*a0*a1^4*a5^3*x1^2
- 8/125*a0^2*a1^2*a2*a5^3*x1^2 + 2/25*a0^3*a2^2*a5^3*x1^2

sage: quintic.theta_covariant(as_form=True)
Binary quadratic given by ...

class sage.rings.invariants.invariant_theory.FormsBase(n, homogeneous, ring, variables)#

Bases: SageObject

The common base class of AlgebraicForm and SeveralAlgebraicForms.

This is an abstract base class to provide common methods. It does not make much sense to instantiate it.

is_homogeneous()#

Return whether the forms were defined by homogeneous polynomials.

OUTPUT:

Boolean. Whether the user originally defined the form via homogeneous variables.

EXAMPLES:

sage: R.<x,y,t> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+y^4+t*x^2*y^2, [x,y])
sage: quartic.is_homogeneous()
True
sage: quartic.form()
x^2*y^2*t + x^4 + y^4

sage: R.<x,y,t> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+1+t*x^2, [x])
sage: quartic.is_homogeneous()
False
sage: quartic.form()
x^4 + x^2*t + 1

ring()#

Return the polynomial ring.

OUTPUT:

A polynomial ring. This is where the defining polynomial(s) live. Note that the polynomials may be homogeneous or inhomogeneous, depending on how the user constructed the object.

EXAMPLES:

sage: R.<x,y,t> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+y^4+t*x^2*y^2, [x,y])
sage: quartic.ring()
Multivariate Polynomial Ring in x, y, t over Rational Field

sage: R.<x,y,t> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+1+t*x^2, [x])
sage: quartic.ring()
Multivariate Polynomial Ring in x, y, t over Rational Field

variables()#

Return the variables of the form.

OUTPUT:

A tuple of variables. If inhomogeneous notation is used for the defining polynomial then the last entry will be None.

EXAMPLES:

sage: R.<x,y,t> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+y^4+t*x^2*y^2, [x,y])
sage: quartic.variables()
(x, y)

sage: R.<x,y,t> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+1+t*x^2, [x])
sage: quartic.variables()
(x, None)

class sage.rings.invariants.invariant_theory.InvariantTheoryFactory#

Bases: object

Factory object for invariants of multilinear forms.

Use the invariant_theory object to construct algebraic forms. These can then be queried for invariant and covariants.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: invariant_theory.ternary_cubic(x^3+y^3+z^3)
Ternary cubic with coefficients (1, 1, 1, 0, 0, 0, 0, 0, 0, 0)

sage: invariant_theory.ternary_cubic(x^3+y^3+z^3).J_covariant()
x^6*y^3 - x^3*y^6 - x^6*z^3 + y^6*z^3 + x^3*z^6 - y^3*z^6

binary_form_from_invariants(degree, invariants, variables=None, as_form=True, *args, **kwargs)#

Reconstruct a binary form from the values of its invariants.

INPUT:

degree – The degree of the binary form.
invariants – A list or tuple of values of the invariants of the binary form.
variables – A list or tuple of two variables that are used for the resulting form (only if as_form is True). If no variables are provided, two abstract variables x and z will be used.
as_form – boolean. If False, the function will return a tuple of coefficients of a binary form.

OUTPUT:

A binary form or a tuple of its coefficients, whose invariants are equal to the given invariants up to a scaling.

EXAMPLES:

In the case of binary quadratics and cubics, the form is reconstructed based on the value of the discriminant. See also binary_quadratic_coefficients_from_invariants() and binary_cubic_coefficients_from_invariants(). These methods will always return the same result if the discriminant is non-zero:

sage: discriminant = 1
sage: invariant_theory.binary_form_from_invariants(2, [discriminant])
Binary quadratic with coefficients (1, -1/4, 0)
sage: invariant_theory.binary_form_from_invariants(3, [discriminant], as_form=false)
(0, 1, -1, 0)

For binary cubics, there is no class implemented yet, so as_form=True will yield an NotImplementedError:

sage: invariant_theory.binary_form_from_invariants(3, [discriminant])
Traceback (most recent call last):
...
NotImplementedError: no class for binary cubics implemented

For binary quintics, the three Clebsch invariants of the form should be provided to reconstruct the form. For more details about these invariants, see clebsch_invariants():

sage: invariants = [1, 0, 0]
sage: invariant_theory.binary_form_from_invariants(5, invariants)
Binary quintic with coefficients (1, 0, 0, 0, 0, 1)

An optional scaling argument may be provided in order to scale the resulting quintic. For more details, see binary_quintic_coefficients_from_invariants():

sage: invariants = [3, 4, 7]
sage: invariant_theory.binary_form_from_invariants(5, invariants)
Binary quintic with coefficients (-37725479487783/1048576,
565882192316745/8388608, 0, 1033866765362693115/67108864,
12849486940936328715/268435456, -23129076493685391687/2147483648)
sage: invariant_theory.binary_form_from_invariants(5, invariants, scaling='normalized')
Binary quintic with coefficients (24389/892616806656,
4205/11019960576, 0, 1015/209952, -145/1296, -3/16)
sage: invariant_theory.binary_form_from_invariants(5, invariants, scaling='coprime')
Binary quintic with coefficients (-2048, 3840, 0, 876960, 2724840,
-613089)

The invariants can also be computed using the invariants of a given binary quintic. The resulting form has the same invariants up to scaling, is \(GL(2,\QQ)\)-equivalent to the provided form and hence has the same canonical form (see canonical_form()):

sage: R.<x0, x1> = QQ[]
sage: p = 3*x1^5 + 6*x1^4*x0 + 3*x1^3*x0^2 + 4*x1^2*x0^3 - 5*x1*x0^4 + 4*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: invariants = quintic.clebsch_invariants(as_tuple=True)
sage: newquintic = invariant_theory.binary_form_from_invariants(5, invariants, variables=quintic.variables())
sage: newquintic
Binary quintic with coefficients (9592267437341790539005557/244140625000000,
2149296928207625556323004064707/610351562500000000,
11149651890347700974453304786783/76293945312500000,
122650775751894638395648891202734239/47683715820312500000,
323996630945706528474286334593218447/11920928955078125000,
1504506503644608395841632538558481466127/14901161193847656250000)
sage: quintic.canonical_form() == newquintic.canonical_form()
True

For binary forms of other degrees, no reconstruction has been implemented yet. For forms of degree 6, see trac ticket #26462:

sage: invariant_theory.binary_form_from_invariants(6, invariants)
Traceback (most recent call last):
...
NotImplementedError: no reconstruction for binary forms of degree 6 implemented

binary_quadratic(quadratic, *args)#

Invariant theory of a quadratic in two variables.

INPUT:

quadratic – a quadratic form.
x, y – the homogeneous variables. If y is None, the quadratic is assumed to be inhomogeneous.

REFERENCES:

Wikipedia article Invariant_of_a_binary_form

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: invariant_theory.binary_quadratic(x^2+y^2)
Binary quadratic with coefficients (1, 1, 0)

sage: T.<t> = QQ[]
sage: invariant_theory.binary_quadratic(t^2 + 2*t + 1, [t])
Binary quadratic with coefficients (1, 1, 2)

binary_quartic(quartic, *args, **kwds)#

Invariant theory of a quartic in two variables.

The algebra of invariants of a quartic form is generated by invariants \(i\), \(j\) of degrees 2, 3. This ring is naturally isomorphic to the ring of modular forms of level 1, with the two generators corresponding to the Eisenstein series \(E_4\) (see EisensteinD()) and \(E_6\) (see EisensteinE()). The algebra of covariants is generated by these two invariants together with the form \(f\) of degree 1 and order 4, the Hessian \(g\) (see g_covariant()) of degree 2 and order 4, and a covariant \(h\) (see h_covariant()) of degree 3 and order 6. They are related by a syzygy

\[j f^3 - g f^2 i + 4 g^3 + h^2 = 0\]

of degree 6 and order 12.

INPUT:

quartic – a quartic.
x, y – the homogeneous variables. If y is None, the quartic is assumed to be inhomogeneous.

REFERENCES:

Wikipedia article Invariant_of_a_binary_form

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: quartic = invariant_theory.binary_quartic(x^4+y^4)
sage: quartic
Binary quartic with coefficients (1, 0, 0, 0, 1)
sage: type(quartic)
<class 'sage.rings.invariants.invariant_theory.BinaryQuartic'>

binary_quintic(quintic, *args, **kwds)#

Create a binary quintic for computing invariants.

A binary quintic is a homogeneous polynomial of degree 5 in two variables. The algebra of invariants of a binary quintic is generated by the invariants \(A\), \(B\) and \(C\) of respective degrees 4, 8 and 12 (see A_invariant(), B_invariant() and C_invariant()).

INPUT:

quintic – a homogeneous polynomial of degree five in two variables or a (possibly inhomogeneous) polynomial of degree at most five in one variable.
*args – the two homogeneous variables. If only one variable is given, the polynomial quintic is assumed to be univariate. If no variables are given, they are guessed.

REFERENCES:

EXAMPLES:

If no variables are provided, they will be guessed:

sage: R.<x,y> = QQ[]
sage: quintic = invariant_theory.binary_quintic(x^5+y^5)
sage: quintic
Binary quintic with coefficients (1, 0, 0, 0, 0, 1)

If only one variable is given, the quintic is the homogenisation of the provided polynomial:

sage: quintic = invariant_theory.binary_quintic(x^5+y^5, x)
sage: quintic
Binary quintic with coefficients (y^5, 0, 0, 0, 0, 1)
sage: quintic.is_homogeneous()
False

If the polynomial has three or more variables, the variables should be specified:

sage: R.<x,y,z> = QQ[]
sage: quintic = invariant_theory.binary_quintic(x^5+z*y^5)
Traceback (most recent call last):
...
ValueError: need 2 or 1 variables, got (x, y, z)
sage: quintic = invariant_theory.binary_quintic(x^5+z*y^5, x, y)
sage: quintic
Binary quintic with coefficients (z, 0, 0, 0, 0, 1)

sage: type(quintic)
<class 'sage.rings.invariants.invariant_theory.BinaryQuintic'>

inhomogeneous_quadratic_form(polynomial, *args)#

Invariants of an inhomogeneous quadratic form.

INPUT:

polynomial – an inhomogeneous quadratic form.
*args – the variables as multiple arguments, or as a single list/tuple.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: quadratic = x^2+2*y^2+3*x*y+4*x+5*y+6
sage: inv3 = invariant_theory.inhomogeneous_quadratic_form(quadratic)
sage: type(inv3)
<class 'sage.rings.invariants.invariant_theory.TernaryQuadratic'>
sage: inv4 = invariant_theory.inhomogeneous_quadratic_form(x^2+y^2+z^2)
sage: type(inv4)
<class 'sage.rings.invariants.invariant_theory.QuadraticForm'>

quadratic_form(polynomial, *args)#

Invariants of a homogeneous quadratic form.

INPUT:

polynomial – a homogeneous or inhomogeneous quadratic form.
*args – the variables as multiple arguments, or as a single list/tuple. If the last argument is None, the cubic is assumed to be inhomogeneous.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: quadratic = x^2+y^2+z^2
sage: inv = invariant_theory.quadratic_form(quadratic)
sage: type(inv)
<class 'sage.rings.invariants.invariant_theory.TernaryQuadratic'>

If some of the ring variables are to be treated as coefficients you need to specify the polynomial variables:

sage: R.<x,y,z, a,b> = QQ[]
sage: quadratic = a*x^2+b*y^2+z^2+2*y*z
sage: invariant_theory.quadratic_form(quadratic, x,y,z)
Ternary quadratic with coefficients (a, b, 1, 0, 0, 2)
sage: invariant_theory.quadratic_form(quadratic, [x,y,z])   # alternate syntax
Ternary quadratic with coefficients (a, b, 1, 0, 0, 2)

Inhomogeneous quadratic forms (see also inhomogeneous_quadratic_form()) can be specified by passing None as the last variable:

sage: inhom = quadratic.subs(z=1)
sage: invariant_theory.quadratic_form(inhom, x,y,None)
Ternary quadratic with coefficients (a, b, 1, 0, 0, 2)

quaternary_biquadratic(quadratic1, quadratic2, *args, **kwds)#

Invariants of two quadratics in four variables.

INPUT:

quadratic1, quadratic2 – two polynomials. Either homogeneous quadratic in 4 homogeneous variables, or inhomogeneous quadratic in 3 variables.
w, x, y, z – the variables. If z is None, the quadratics are assumed to be inhomogeneous.

EXAMPLES:

sage: R.<w,x,y,z> = QQ[]
sage: q1 = w^2+x^2+y^2+z^2
sage: q2 = w*x + y*z
sage: inv = invariant_theory.quaternary_biquadratic(q1, q2)
sage: type(inv)
<class 'sage.rings.invariants.invariant_theory.TwoQuaternaryQuadratics'>

Distance between two spheres [Sal1958], [Sal1965]

sage: R.<x,y,z, a,b,c, r1,r2> = QQ[]
sage: S1 = -r1^2 + x^2 + y^2 + z^2
sage: S2 = -r2^2 + (x-a)^2 + (y-b)^2 + (z-c)^2
sage: inv = invariant_theory.quaternary_biquadratic(S1, S2, [x, y, z])
sage: inv.Delta_invariant()
-r1^2
sage: inv.Delta_prime_invariant()
-r2^2
sage: inv.Theta_invariant()
a^2 + b^2 + c^2 - 3*r1^2 - r2^2
sage: inv.Theta_prime_invariant()
a^2 + b^2 + c^2 - r1^2 - 3*r2^2
sage: inv.Phi_invariant()
2*a^2 + 2*b^2 + 2*c^2 - 3*r1^2 - 3*r2^2
sage: inv.J_covariant()
0

quaternary_quadratic(quadratic, *args)#

Invariant theory of a quadratic in four variables.

INPUT:

quadratic – a quadratic form.
w, x, y, z – the homogeneous variables. If z is None, the quadratic is assumed to be inhomogeneous.

REFERENCES:

Wikipedia article Invariant_of_a_binary_form

EXAMPLES:

sage: R.<w,x,y,z> = QQ[]
sage: invariant_theory.quaternary_quadratic(w^2+x^2+y^2+z^2)
Quaternary quadratic with coefficients (1, 1, 1, 1, 0, 0, 0, 0, 0, 0)

sage: R.<x,y,z> = QQ[]
sage: invariant_theory.quaternary_quadratic(1+x^2+y^2+z^2)
Quaternary quadratic with coefficients (1, 1, 1, 1, 0, 0, 0, 0, 0, 0)

ternary_biquadratic(quadratic1, quadratic2, *args, **kwds)#

Invariants of two quadratics in three variables.

INPUT:

quadratic1, quadratic2 – two polynomials. Either homogeneous quadratic in 3 homogeneous variables, or inhomogeneous quadratic in 2 variables.
x, y, z – the variables. If z is None, the quadratics are assumed to be inhomogeneous.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: q1 = x^2+y^2+z^2
sage: q2 = x*y + y*z + x*z
sage: inv = invariant_theory.ternary_biquadratic(q1, q2)
sage: type(inv)
<class 'sage.rings.invariants.invariant_theory.TwoTernaryQuadratics'>

Distance between two circles:

sage: R.<x,y, a,b, r1,r2> = QQ[]
sage: S1 = -r1^2 + x^2 + y^2
sage: S2 = -r2^2 + (x-a)^2 + (y-b)^2
sage: inv = invariant_theory.ternary_biquadratic(S1, S2, [x, y])
sage: inv.Delta_invariant()
-r1^2
sage: inv.Delta_prime_invariant()
-r2^2
sage: inv.Theta_invariant()
a^2 + b^2 - 2*r1^2 - r2^2
sage: inv.Theta_prime_invariant()
a^2 + b^2 - r1^2 - 2*r2^2
sage: inv.F_covariant()
2*x^2*a^2 + y^2*a^2 - 2*x*a^3 + a^4 + 2*x*y*a*b - 2*y*a^2*b + x^2*b^2 +
2*y^2*b^2 - 2*x*a*b^2 + 2*a^2*b^2 - 2*y*b^3 + b^4 - 2*x^2*r1^2 - 2*y^2*r1^2 +
2*x*a*r1^2 - 2*a^2*r1^2 + 2*y*b*r1^2 - 2*b^2*r1^2 + r1^4 - 2*x^2*r2^2 -
2*y^2*r2^2 + 2*x*a*r2^2 - 2*a^2*r2^2 + 2*y*b*r2^2 - 2*b^2*r2^2 + 2*r1^2*r2^2 +
r2^4
sage: inv.J_covariant()
-8*x^2*y*a^3 + 8*x*y*a^4 + 8*x^3*a^2*b - 16*x*y^2*a^2*b - 8*x^2*a^3*b +
8*y^2*a^3*b + 16*x^2*y*a*b^2 - 8*y^3*a*b^2 + 8*x*y^2*b^3 - 8*x^2*a*b^3 +
8*y^2*a*b^3 - 8*x*y*b^4 + 8*x*y*a^2*r1^2 - 8*y*a^3*r1^2 - 8*x^2*a*b*r1^2 +
8*y^2*a*b*r1^2 + 8*x*a^2*b*r1^2 - 8*x*y*b^2*r1^2 - 8*y*a*b^2*r1^2 + 8*x*b^3*r1^2 -
8*x*y*a^2*r2^2 + 8*x^2*a*b*r2^2 - 8*y^2*a*b*r2^2 + 8*x*y*b^2*r2^2

ternary_cubic(cubic, *args, **kwds)#

Invariants of a cubic in three variables.

The algebra of invariants of a ternary cubic under \(SL_3(\CC)\) is a polynomial algebra generated by two invariants \(S\) (see S_invariant()) and T (see T_invariant()) of degrees 4 and 6, called Aronhold invariants.

The ring of covariants is given as follows. The identity covariant U of a ternary cubic has degree 1 and order 3. The Hessian \(H\) (see Hessian()) is a covariant of ternary cubics of degree 3 and order 3. There is a covariant \(\Theta\) (see Theta_covariant()) of ternary cubics of degree 8 and order 6 that vanishes on points \(x\) lying on the Salmon conic of the polar of \(x\) with respect to the curve and its Hessian curve. The Brioschi covariant \(J\) (see J_covariant()) is the Jacobian of \(U\), \(\Theta\), and \(H\) of degree 12, order 9. The algebra of covariants of a ternary cubic is generated over the ring of invariants by \(U\), \(\Theta\), \(H\), and \(J\), with a relation

\[\begin{split}\begin{split} J^2 =& 4 \Theta^3 + T U^2 \Theta^2 + \Theta (-4 S^3 U^4 + 2 S T U^3 H - 72 S^2 U^2 H^2 \\ & - 18 T U H^3 + 108 S H^4) -16 S^4 U^5 H - 11 S^2 T U^4 H^2 \\ & -4 T^2 U^3 H^3 +54 S T U^2 H^4 -432 S^2 U H^5 -27 T H^6 \end{split}\end{split}\]

REFERENCES:

Wikipedia article Ternary_cubic

INPUT:

cubic – a homogeneous cubic in 3 homogeneous variables, or an inhomogeneous cubic in 2 variables.
x, y, z – the variables. If z is None, the cubic is assumed to be inhomogeneous.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+z^3)
sage: type(cubic)
<class 'sage.rings.invariants.invariant_theory.TernaryCubic'>

ternary_quadratic(quadratic, *args, **kwds)#

Invariants of a quadratic in three variables.

INPUT:

quadratic – a homogeneous quadratic in 3 homogeneous variables, or an inhomogeneous quadratic in 2 variables.
x, y, z – the variables. If z is None, the quadratic is assumed to be inhomogeneous.

REFERENCES:

Wikipedia article Invariant_of_a_binary_form

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: invariant_theory.ternary_quadratic(x^2+y^2+z^2)
Ternary quadratic with coefficients (1, 1, 1, 0, 0, 0)

sage: T.<u, v> = QQ[]
sage: invariant_theory.ternary_quadratic(1+u^2+v^2)
Ternary quadratic with coefficients (1, 1, 1, 0, 0, 0)

sage: quadratic = x^2+y^2+z^2
sage: inv = invariant_theory.ternary_quadratic(quadratic)
sage: type(inv)
<class 'sage.rings.invariants.invariant_theory.TernaryQuadratic'>

class sage.rings.invariants.invariant_theory.QuadraticForm(n, d, polynomial, *args)#

Bases: AlgebraicForm

Invariant theory of a multivariate quadratic form.

You should use the invariant_theory factory object to construct instances of this class. See quadratic_form() for details.

as_QuadraticForm()#

Convert into a QuadraticForm.

OUTPUT:

Sage has a special quadratic forms subsystem. This method converts self into this QuadraticForm representation.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: p = x^2+y^2+z^2+2*x*y+3*x*z
sage: quadratic = invariant_theory.ternary_quadratic(p)
sage: matrix(quadratic)
[  1   1 3/2]
[  1   1   0]
[3/2   0   1]
sage: quadratic.as_QuadraticForm()
Quadratic form in 3 variables over Multivariate Polynomial
Ring in x, y, z over Rational Field with coefficients:
[ 1 2 3 ]
[ * 1 0 ]
[ * * 1 ]
sage: _.polynomial('X,Y,Z')
X^2 + 2*X*Y + Y^2 + 3*X*Z + Z^2

coeffs()#

The coefficients of a quadratic form.

Given

\[f(x) = \sum_{0\leq i<n} a_i x_i^2 + \sum_{0\leq j <k<n} a_{jk} x_j x_k\]

this function returns \(a = (a_0, \dots, a_n, a_{00}, a_{01}, \dots, a_{n-1,n})\)

EXAMPLES:

sage: R.<a,b,c,d,e,f,g, x,y,z> = QQ[]
sage: p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z
sage: inv = invariant_theory.quadratic_form(p, x,y,z); inv
Ternary quadratic with coefficients (a, b, c, d, e, f)
sage: inv.coeffs()
(a, b, c, d, e, f)
sage: inv.scaled_coeffs()
(a, b, c, 1/2*d, 1/2*e, 1/2*f)

discriminant()#

Return the discriminant of the quadratic form.

Up to an overall constant factor, this is just the determinant of the defining matrix, see matrix(). For a quadratic form in \(n\) variables, the overall constant is \(2^{n-1}\) if \(n\) is odd and \((-1)^{n/2} 2^n\) if \(n\) is even.

EXAMPLES:

sage: R.<a,b,c, x,y> = QQ[]
sage: p = a*x^2+b*x*y+c*y^2
sage: quadratic = invariant_theory.quadratic_form(p, x,y)
sage: quadratic.discriminant()
b^2 - 4*a*c

sage: R.<a,b,c,d,e,f,g, x,y,z> = QQ[]
sage: p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z
sage: quadratic = invariant_theory.quadratic_form(p, x,y,z)
sage: quadratic.discriminant()
4*a*b*c - c*d^2 - b*e^2 + d*e*f - a*f^2

dual()#

Return the dual quadratic form.

OUTPUT:

A new quadratic form (with the same number of variables) defined by the adjoint matrix.

EXAMPLES:

sage: R.<a,b,c,x,y,z> = QQ[]
sage: cubic = x^2+y^2+z^2
sage: quadratic = invariant_theory.ternary_quadratic(a*x^2+b*y^2+c*z^2, [x,y,z])
sage: quadratic.form()
a*x^2 + b*y^2 + c*z^2
sage: quadratic.dual().form()
b*c*x^2 + a*c*y^2 + a*b*z^2

sage: R.<x,y,z, t> = QQ[]
sage: cubic = x^2+y^2+z^2
sage: quadratic = invariant_theory.ternary_quadratic(x^2+y^2+z^2 + t*x*y, [x,y,z])
sage: quadratic.dual()
Ternary quadratic with coefficients (1, 1, -1/4*t^2 + 1, -t, 0, 0)

sage: R.<x,y, t> = QQ[]
sage: quadratic = invariant_theory.ternary_quadratic(x^2+y^2+1 + t*x*y, [x,y])
sage: quadratic.dual()
Ternary quadratic with coefficients (1, 1, -1/4*t^2 + 1, -t, 0, 0)

classmethod from_invariants(discriminant, x, z, *args, **kwargs)#

Construct a binary quadratic from its discriminant.

This function constructs a binary quadratic whose discriminant equal the one provided as argument up to scaling.

INPUT:

discriminant – Value of the discriminant used to reconstruct the binary quadratic.

OUTPUT:

A QuadraticForm with 2 variables.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: from sage.rings.invariants.invariant_theory import QuadraticForm
sage: QuadraticForm.from_invariants(1, x, y)
Binary quadratic with coefficients (1, -1/4, 0)

invariants(type='discriminant')#

Return a tuple of invariants of a binary quadratic.

INPUT:

type – The type of invariants to return. The default choice is to return the discriminant.

OUTPUT:

The invariants of the binary quadratic.

EXAMPLES:

sage: R.<x0, x1> = QQ[]
sage: p = 2*x1^2 + 5*x0*x1 + 3*x0^2
sage: quadratic = invariant_theory.binary_quadratic(p, x0, x1)
sage: quadratic.invariants()
(1,)
sage: quadratic.invariants('unknown')
Traceback (most recent call last):
...
ValueError: unknown type of invariants unknown for a binary quadratic

matrix()#

Return the quadratic form as a symmetric matrix

OUTPUT:

This method returns a symmetric matrix \(A\) such that the quadratic \(Q\) equals

\[Q(x,y,z,\dots) = (x,y,\dots) A (x,y,\dots)^t\]

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: quadratic = invariant_theory.ternary_quadratic(x^2+y^2+z^2+x*y)
sage: matrix(quadratic)
[  1 1/2   0]
[1/2   1   0]
[  0   0   1]
sage: quadratic._matrix_() == matrix(quadratic)
True

monomials()#

List the basis monomials in the form.

OUTPUT:

A tuple of monomials. They are in the same order as coeffs().

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: quadratic = invariant_theory.quadratic_form(x^2+y^2)
sage: quadratic.monomials()
(x^2, y^2, x*y)

sage: quadratic = invariant_theory.inhomogeneous_quadratic_form(x^2+y^2)
sage: quadratic.monomials()
(x^2, y^2, 1, x*y, x, y)

scaled_coeffs()#

The scaled coefficients of a quadratic form.

Given

\[f(x) = \sum_{0\leq i<n} a_i x_i^2 + \sum_{0\leq j <k<n} 2 a_{jk} x_j x_k\]

this function returns \(a = (a_0, \cdots, a_n, a_{00}, a_{01}, \dots, a_{n-1,n})\)

EXAMPLES:

sage: R.<a,b,c,d,e,f,g, x,y,z> = QQ[]
sage: p = a*x^2 + b*y^2 + c*z^2 + d*x*y + e*x*z + f*y*z
sage: inv = invariant_theory.quadratic_form(p, x,y,z); inv
Ternary quadratic with coefficients (a, b, c, d, e, f)
sage: inv.coeffs()
(a, b, c, d, e, f)
sage: inv.scaled_coeffs()
(a, b, c, 1/2*d, 1/2*e, 1/2*f)

class sage.rings.invariants.invariant_theory.SeveralAlgebraicForms(forms)#

Bases: FormsBase

The base class of multiple algebraic forms (i.e. homogeneous polynomials).

You should only instantiate the derived classes of this base class.

See AlgebraicForm for the base class of a single algebraic form.

INPUT:

forms – a list/tuple/iterable of at least one AlgebraicForm object, all with the same number of variables. Interpreted as multiple homogeneous polynomials in a common polynomial ring.

EXAMPLES:

sage: from sage.rings.invariants.invariant_theory import AlgebraicForm, SeveralAlgebraicForms
sage: R.<x,y> = QQ[]
sage: p = AlgebraicForm(2, 2, x^2, (x,y))
sage: q = AlgebraicForm(2, 2, y^2, (x,y))
sage: pq = SeveralAlgebraicForms([p, q])

get_form(i)#

Return the \(i\)-th form.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: q1 = invariant_theory.quadratic_form(x^2 + y^2)
sage: q2 = invariant_theory.quadratic_form(x*y)
sage: from sage.rings.invariants.invariant_theory import SeveralAlgebraicForms
sage: q12 = SeveralAlgebraicForms([q1, q2])
sage: q12.get_form(0) is q1
True
sage: q12.get_form(1) is q2
True
sage: q12[0] is q12.get_form(0)   # syntactic sugar
True
sage: q12[1] is q12.get_form(1)   # syntactic sugar
True

homogenized(var='h')#

Return form as defined by a homogeneous polynomial.

INPUT:

var – either a variable name, variable index or a variable (default: 'h').

OUTPUT:

The same algebraic form, but defined by a homogeneous polynomial.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: q = invariant_theory.quaternary_biquadratic(x^2+1, y^2+1, [x,y,z])
sage: q
Joint quaternary quadratic with coefficients (1, 0, 0, 1, 0, 0, 0, 0, 0, 0)
and quaternary quadratic with coefficients (0, 1, 0, 1, 0, 0, 0, 0, 0, 0)
sage: q.homogenized()
Joint quaternary quadratic with coefficients (1, 0, 0, 1, 0, 0, 0, 0, 0, 0)
and quaternary quadratic with coefficients (0, 1, 0, 1, 0, 0, 0, 0, 0, 0)
sage: type(q) is type(q.homogenized())
True

n_forms()#

Return the number of forms.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: q1 = invariant_theory.quadratic_form(x^2 + y^2)
sage: q2 = invariant_theory.quadratic_form(x*y)
sage: from sage.rings.invariants.invariant_theory import SeveralAlgebraicForms
sage: q12 = SeveralAlgebraicForms([q1, q2])
sage: q12.n_forms()
2
sage: len(q12) == q12.n_forms()    # syntactic sugar
True

class sage.rings.invariants.invariant_theory.TernaryCubic(n, d, polynomial, *args)#

Bases: AlgebraicForm

Invariant theory of a ternary cubic.

You should use the invariant_theory factory object to construct instances of this class. See ternary_cubic() for details.

Hessian()#

Return the Hessian covariant.

OUTPUT:

The Hessian matrix multiplied with the conventional normalization factor \(1/216\).

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+z^3)
sage: cubic.Hessian()
x*y*z

sage: R.<x,y> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+1)
sage: cubic.Hessian()
x*y

J_covariant()#

Return the J-covariant of the ternary cubic.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+z^3)
sage: cubic.J_covariant()
x^6*y^3 - x^3*y^6 - x^6*z^3 + y^6*z^3 + x^3*z^6 - y^3*z^6

sage: R.<x,y> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+1)
sage: cubic.J_covariant()
x^6*y^3 - x^3*y^6 - x^6 + y^6 + x^3 - y^3

S_invariant()#

Return the S-invariant.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^2*y+y^3+z^3+x*y*z)
sage: cubic.S_invariant()
-1/1296

T_invariant()#

Return the T-invariant.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+z^3)
sage: cubic.T_invariant()
1

sage: R.<x,y,z,t> = GF(7)[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+z^3+t*x*y*z, [x,y,z])
sage: cubic.T_invariant()
-t^6 - t^3 + 1

Theta_covariant()#

Return the \(\Theta\) covariant.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+z^3)
sage: cubic.Theta_covariant()
-x^3*y^3 - x^3*z^3 - y^3*z^3

sage: R.<x,y> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y^3+1)
sage: cubic.Theta_covariant()
-x^3*y^3 - x^3 - y^3

sage: R.<x,y,z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[]
sage: p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z +
....:       a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 )
sage: cubic = invariant_theory.ternary_cubic(p, x,y,z)
sage: len(list(cubic.Theta_covariant()))
6952

coeffs()#

Return the coefficients of a cubic.

Given

\[\begin{split}\begin{split} p(x,y) =&\; a_{30} x^{3} + a_{21} x^{2} y + a_{12} x y^{2} + a_{03} y^{3} + a_{20} x^{2} + \\ &\; a_{11} x y + a_{02} y^{2} + a_{10} x + a_{01} y + a_{00} \end{split}\end{split}\]

this function returns \(a = (a_{30}, a_{03}, a_{00}, a_{21}, a_{20}, a_{12}, a_{02}, a_{10}, a_{01}, a_{11})\)

EXAMPLES:

sage: R.<x,y,z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[]
sage: p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z +
....:       a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 )
sage: invariant_theory.ternary_cubic(p, x,y,z).coeffs()
(a30, a03, a00, a21, a20, a12, a02, a10, a01, a11)
sage: invariant_theory.ternary_cubic(p.subs(z=1), x, y).coeffs()
(a30, a03, a00, a21, a20, a12, a02, a10, a01, a11)

monomials()#

List the basis monomials of the form.

OUTPUT:

A tuple of monomials. They are in the same order as coeffs().

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: cubic = invariant_theory.ternary_cubic(x^3+y*z^2)
sage: cubic.monomials()
(x^3, y^3, z^3, x^2*y, x^2*z, x*y^2, y^2*z, x*z^2, y*z^2, x*y*z)

polar_conic()#

Return the polar conic of the cubic.

OUTPUT:

Given the ternary cubic \(f(X,Y,Z)\), this method returns the symmetric matrix \(A(x,y,z)\) defined by

\[x f_X + y f_Y + z f_Z = (X,Y,Z) \cdot A(x,y,z) \cdot (X,Y,Z)^t\]

EXAMPLES:

sage: R.<x,y,z,X,Y,Z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[]
sage: p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z +
....:       a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 )
sage: cubic = invariant_theory.ternary_cubic(p, x,y,z)
sage: cubic.polar_conic()
[  3*x*a30 + y*a21 + z*a20 x*a21 + y*a12 + 1/2*z*a11 x*a20 + 1/2*y*a11 + z*a10]
[x*a21 + y*a12 + 1/2*z*a11   x*a12 + 3*y*a03 + z*a02 1/2*x*a11 + y*a02 + z*a01]
[x*a20 + 1/2*y*a11 + z*a10 1/2*x*a11 + y*a02 + z*a01   x*a10 + y*a01 + 3*z*a00]

sage: polar_eqn = X*p.derivative(x) + Y*p.derivative(y) + Z*p.derivative(z)
sage: polar = invariant_theory.ternary_quadratic(polar_eqn, [x,y,z])
sage: polar.matrix().subs(X=x,Y=y,Z=z) == cubic.polar_conic()
True

scaled_coeffs()#

Return the coefficients of a cubic.

Compared to coeffs(), this method returns rescaled coefficients that are often used in invariant theory.

Given

this function returns \(a = (a_{30}, a_{03}, a_{00}, a_{21}/3, a_{20}/3, a_{12}/3, a_{02}/3, a_{10}/3, a_{01}/3, a_{11}/6)\)

EXAMPLES:

sage: R.<x,y,z,a30,a21,a12,a03,a20,a11,a02,a10,a01,a00> = QQ[]
sage: p = ( a30*x^3 + a21*x^2*y + a12*x*y^2 + a03*y^3 + a20*x^2*z +
....:       a11*x*y*z + a02*y^2*z + a10*x*z^2 + a01*y*z^2 + a00*z^3 )
sage: invariant_theory.ternary_cubic(p, x,y,z).scaled_coeffs()
(a30, a03, a00, 1/3*a21, 1/3*a20, 1/3*a12, 1/3*a02, 1/3*a10, 1/3*a01, 1/6*a11)

syzygy(U, S, T, H, Theta, J)#

Return the syzygy of the cubic evaluated on the invariants and covariants.

INPUT:

U, S, T, H, Theta, J – polynomials from the same polynomial ring.

OUTPUT:

0 if evaluated for the form, the S invariant, the T invariant, the Hessian, the \(\Theta\) covariant and the J-covariant of a ternary cubic.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: monomials = (x^3, y^3, z^3, x^2*y, x^2*z, x*y^2,
....:              y^2*z, x*z^2, y*z^2, x*y*z)
sage: random_poly = sum([ randint(0,10000) * m for m in monomials ])
sage: cubic = invariant_theory.ternary_cubic(random_poly)
sage: U = cubic.form()
sage: S = cubic.S_invariant()
sage: T = cubic.T_invariant()
sage: H = cubic.Hessian()
sage: Theta = cubic.Theta_covariant()
sage: J = cubic.J_covariant()
sage: cubic.syzygy(U, S, T, H, Theta, J)
0

class sage.rings.invariants.invariant_theory.TernaryQuadratic(n, d, polynomial, *args)#

Bases: QuadraticForm

Invariant theory of a ternary quadratic.

You should use the invariant_theory factory object to construct instances of this class. See ternary_quadratic() for details.

coeffs()#

Return the coefficients of a quadratic.

Given

\[p(x,y) =&\; a_{20} x^{2} + a_{11} x y + a_{02} y^{2} + a_{10} x + a_{01} y + a_{00}\]

this function returns \(a = (a_{20}, a_{02}, a_{00}, a_{11}, a_{10}, a_{01} )\)

EXAMPLES:

sage: R.<x,y,z,a20,a11,a02,a10,a01,a00> = QQ[]
sage: p = ( a20*x^2 + a11*x*y + a02*y^2 +
....:       a10*x*z + a01*y*z + a00*z^2 )
sage: invariant_theory.ternary_quadratic(p, x,y,z).coeffs()
(a20, a02, a00, a11, a10, a01)
sage: invariant_theory.ternary_quadratic(p.subs(z=1), x, y).coeffs()
(a20, a02, a00, a11, a10, a01)

covariant_conic(other)#

Return the ternary quadratic covariant to self and other.

INPUT:

other – Another ternary quadratic.

OUTPUT:

The so-called covariant conic, a ternary quadratic. It is symmetric under exchange of self and other.

EXAMPLES:

sage: ring.<x,y,z> = QQ[]
sage: Q = invariant_theory.ternary_quadratic(x^2+y^2+z^2)
sage: R = invariant_theory.ternary_quadratic(x*y+x*z+y*z)
sage: Q.covariant_conic(R)
-x*y - x*z - y*z
sage: R.covariant_conic(Q)
-x*y - x*z - y*z

monomials()#

List the basis monomials of the form.

OUTPUT:

A tuple of monomials. They are in the same order as coeffs().

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: quadratic = invariant_theory.ternary_quadratic(x^2+y*z)
sage: quadratic.monomials()
(x^2, y^2, z^2, x*y, x*z, y*z)

scaled_coeffs()#

Return the scaled coefficients of a quadratic.

Given

\[p(x,y) =&\; a_{20} x^{2} + a_{11} x y + a_{02} y^{2} + a_{10} x + a_{01} y + a_{00}\]

this function returns \(a = (a_{20}, a_{02}, a_{00}, a_{11}/2, a_{10}/2, a_{01}/2, )\)

EXAMPLES:

sage: R.<x,y,z,a20,a11,a02,a10,a01,a00> = QQ[]
sage: p = ( a20*x^2 + a11*x*y + a02*y^2 +
....:       a10*x*z + a01*y*z + a00*z^2 )
sage: invariant_theory.ternary_quadratic(p, x,y,z).scaled_coeffs()
(a20, a02, a00, 1/2*a11, 1/2*a10, 1/2*a01)
sage: invariant_theory.ternary_quadratic(p.subs(z=1), x, y).scaled_coeffs()
(a20, a02, a00, 1/2*a11, 1/2*a10, 1/2*a01)

class sage.rings.invariants.invariant_theory.TwoAlgebraicForms(forms)#

Bases: SeveralAlgebraicForms

first()#

Return the first of the two forms.

OUTPUT:

The first algebraic form used in the definition.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: q0 = invariant_theory.quadratic_form(x^2 + y^2)
sage: q1 = invariant_theory.quadratic_form(x*y)
sage: from sage.rings.invariants.invariant_theory import TwoAlgebraicForms
sage: q = TwoAlgebraicForms([q0, q1])
sage: q.first() is q0
True
sage: q.get_form(0) is q0
True
sage: q.first().polynomial()
x^2 + y^2

second()#

Return the second of the two forms.

OUTPUT:

The second form used in the definition.

EXAMPLES:

sage: R.<x,y> = QQ[]
sage: q0 = invariant_theory.quadratic_form(x^2 + y^2)
sage: q1 = invariant_theory.quadratic_form(x*y)
sage: from sage.rings.invariants.invariant_theory import TwoAlgebraicForms
sage: q = TwoAlgebraicForms([q0, q1])
sage: q.second() is q1
True
sage: q.get_form(1) is q1
True
sage: q.second().polynomial()
x*y

class sage.rings.invariants.invariant_theory.TwoQuaternaryQuadratics(forms)#

Bases: TwoAlgebraicForms

Invariant theory of two quaternary quadratics.

You should use the invariant_theory factory object to construct instances of this class. See quaternary_biquadratics() for details.

REFERENCES:

section on “Invariants and Covariants of Systems of Quadrics” in [Sal1958], [Sal1965]

Delta_invariant()#

Return the \(\Delta\) invariant.

EXAMPLES:

sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3
sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3
sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z])
sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False)
sage: q.Delta_invariant() == coeffs[4]
True

Delta_prime_invariant()#

Return the \(\Delta'\) invariant.

EXAMPLES:

sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3
sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3
sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z])
sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False)
sage: q.Delta_prime_invariant() == coeffs[0]
True

J_covariant()#

The \(J\)-covariant.

This is the Jacobian determinant of the two biquadratics, the \(T\)-covariant, and the \(T'\)-covariant with respect to the four homogeneous variables.

EXAMPLES:

sage: R.<w,x,y,z,a0,a1,a2,a3,A0,A1,A2,A3> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3*w^2
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3*w^2
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [w, x, y, z])
sage: q.J_covariant().factor()
z * y * x * w * (a3*A2 - a2*A3) * (a3*A1 - a1*A3) * (-a2*A1 + a1*A2)
* (a3*A0 - a0*A3) * (-a2*A0 + a0*A2) * (-a1*A0 + a0*A1)

Phi_invariant()#

Return the \(\Phi'\) invariant.

EXAMPLES:

sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3
sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3
sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z])
sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False)
sage: q.Phi_invariant() == coeffs[2]
True

T_covariant()#

The \(T\)-covariant.

EXAMPLES:

sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3
sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3
sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z])
sage: T = invariant_theory.quaternary_quadratic(q.T_covariant(), [x,y,z]).matrix()
sage: M = q[0].matrix().adjugate() + t*q[1].matrix().adjugate()
sage: M = M.adjugate().apply_map(             # long time (4s on my thinkpad W530)
....:         lambda m: m.coefficient(t))
sage: M == q.Delta_invariant()*T             # long time
True

T_prime_covariant()#

The \(T'\)-covariant.

EXAMPLES:

sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3
sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3
sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z])
sage: Tprime = invariant_theory.quaternary_quadratic(
....:     q.T_prime_covariant(), [x,y,z]).matrix()
sage: M = q[0].matrix().adjugate() + t*q[1].matrix().adjugate()
sage: M = M.adjugate().apply_map(                # long time (4s on my thinkpad W530)
....:         lambda m: m.coefficient(t^2))
sage: M == q.Delta_prime_invariant() * Tprime   # long time
True

Theta_invariant()#

Return the \(\Theta\) invariant.

EXAMPLES:

sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3
sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3
sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z])
sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False)
sage: q.Theta_invariant() == coeffs[3]
True

Theta_prime_invariant()#

Return the \(\Theta'\) invariant.

EXAMPLES:

sage: R.<x,y,z,t,a0,a1,a2,a3,b0,b1,b2,b3,b4,b5,A0,A1,A2,A3,B0,B1,B2,B3,B4,B5> = QQ[]
sage: p1 = a0*x^2 + a1*y^2 + a2*z^2 + a3
sage: p1 += b0*x*y + b1*x*z + b2*x + b3*y*z + b4*y + b5*z
sage: p2 = A0*x^2 + A1*y^2 + A2*z^2 + A3
sage: p2 += B0*x*y + B1*x*z + B2*x + B3*y*z + B4*y + B5*z
sage: q = invariant_theory.quaternary_biquadratic(p1, p2, [x, y, z])
sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False)
sage: q.Theta_prime_invariant() == coeffs[1]
True

syzygy(Delta, Theta, Phi, Theta_prime, Delta_prime, U, V, T, T_prime, J)#

Return the syzygy evaluated on the invariants and covariants.

INPUT:

Delta, Theta, Phi, Theta_prime, Delta_prime, U, V, T, T_prime, J – polynomials from the same polynomial ring.

OUTPUT:

Zero if the U is the first polynomial, V the second polynomial, and the remaining input are the invariants and covariants of a quaternary biquadratic.

EXAMPLES:

sage: R.<w,x,y,z> = QQ[]
sage: monomials = [x^2, x*y, y^2, x*z, y*z, z^2, x*w, y*w, z*w, w^2]
sage: def q_rnd():  return sum(randint(-1000,1000)*m for m in monomials)
sage: biquadratic = invariant_theory.quaternary_biquadratic(q_rnd(), q_rnd())
sage: Delta = biquadratic.Delta_invariant()
sage: Theta = biquadratic.Theta_invariant()
sage: Phi = biquadratic.Phi_invariant()
sage: Theta_prime = biquadratic.Theta_prime_invariant()
sage: Delta_prime = biquadratic.Delta_prime_invariant()
sage: U = biquadratic.first().polynomial()
sage: V  = biquadratic.second().polynomial()
sage: T = biquadratic.T_covariant()
sage: T_prime  = biquadratic.T_prime_covariant()
sage: J = biquadratic.J_covariant()
sage: biquadratic.syzygy(Delta, Theta, Phi, Theta_prime, Delta_prime, U, V, T, T_prime, J)
0

If the arguments are not the invariants and covariants then the output is some (generically non-zero) polynomial:

sage: biquadratic.syzygy(1, 1, 1, 1, 1, 1, 1, 1, 1, x)
-x^2 + 1

class sage.rings.invariants.invariant_theory.TwoTernaryQuadratics(forms)#

Bases: TwoAlgebraicForms

Invariant theory of two ternary quadratics.

You should use the invariant_theory factory object to construct instances of this class. See ternary_biquadratics() for details.

REFERENCES:

Section on “Invariants and Covariants of Systems of Conics”, Art. 388 (a) in [Sal1954]

Delta_invariant()#

Return the \(\Delta\) invariant.

EXAMPLES:

sage: R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t> = QQ[]
sage: p1 = a00*y0^2 + 2*a01*y0*y1 + a11*y1^2 + 2*a02*y0*y2 + 2*a12*y1*y2 + a22*y2^2
sage: p2 = b00*y0^2 + 2*b01*y0*y1 + b11*y1^2 + 2*b02*y0*y2 + 2*b12*y1*y2 + b22*y2^2
sage: q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2])
sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False)
sage: q.Delta_invariant() == coeffs[3]
True

Delta_prime_invariant()#

Return the \(\Delta'\) invariant.

EXAMPLES:

sage: R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t> = QQ[]
sage: p1 = a00*y0^2 + 2*a01*y0*y1 + a11*y1^2 + 2*a02*y0*y2 + 2*a12*y1*y2 + a22*y2^2
sage: p2 = b00*y0^2 + 2*b01*y0*y1 + b11*y1^2 + 2*b02*y0*y2 + 2*b12*y1*y2 + b22*y2^2
sage: q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2])
sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False)
sage: q.Delta_prime_invariant() == coeffs[0]
True

F_covariant()#

Return the \(F\) covariant.

EXAMPLES:

sage: R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, x, y> = QQ[]
sage: p1 = 73*x^2 + 96*x*y - 11*y^2 + 4*x + 63*y + 57
sage: p2 = 61*x^2 - 100*x*y - 72*y^2 - 81*x + 39*y - 7
sage: q = invariant_theory.ternary_biquadratic(p1, p2, [x, y])
sage: q.F_covariant()
-32566577*x^2 + 29060637/2*x*y + 20153633/4*y^2 -
30250497/2*x - 241241273/4*y - 323820473/16

J_covariant()#

Return the \(J\) covariant.

EXAMPLES:

sage: R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, x, y> = QQ[]
sage: p1 = 73*x^2 + 96*x*y - 11*y^2 + 4*x + 63*y + 57
sage: p2 = 61*x^2 - 100*x*y - 72*y^2 - 81*x + 39*y - 7
sage: q = invariant_theory.ternary_biquadratic(p1, p2, [x, y])
sage: q.J_covariant()
1057324024445*x^3 + 1209531088209*x^2*y + 942116599708*x*y^2 +
984553030871*y^3 + 543715345505/2*x^2 - 3065093506021/2*x*y +
755263948570*y^2 - 1118430692650*x - 509948695327/4*y + 3369951531745/8

Theta_invariant()#

Return the \(\Theta\) invariant.

EXAMPLES:

sage: R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t> = QQ[]
sage: p1 = a00*y0^2 + 2*a01*y0*y1 + a11*y1^2 + 2*a02*y0*y2 + 2*a12*y1*y2 + a22*y2^2
sage: p2 = b00*y0^2 + 2*b01*y0*y1 + b11*y1^2 + 2*b02*y0*y2 + 2*b12*y1*y2 + b22*y2^2
sage: q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2])
sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False)
sage: q.Theta_invariant() == coeffs[2]
True

Theta_prime_invariant()#

Return the \(\Theta'\) invariant.

EXAMPLES:

sage: R.<a00, a01, a11, a02, a12, a22, b00, b01, b11, b02, b12, b22, y0, y1, y2, t> = QQ[]
sage: p1 = a00*y0^2 + 2*a01*y0*y1 + a11*y1^2 + 2*a02*y0*y2 + 2*a12*y1*y2 + a22*y2^2
sage: p2 = b00*y0^2 + 2*b01*y0*y1 + b11*y1^2 + 2*b02*y0*y2 + 2*b12*y1*y2 + b22*y2^2
sage: q = invariant_theory.ternary_biquadratic(p1, p2, [y0, y1, y2])
sage: coeffs = det(t * q[0].matrix() + q[1].matrix()).polynomial(t).coefficients(sparse=False)
sage: q.Theta_prime_invariant() == coeffs[1]
True

syzygy(Delta, Theta, Theta_prime, Delta_prime, S, S_prime, F, J)#

Return the syzygy evaluated on the invariants and covariants.

INPUT:

Delta, Theta, Theta_prime, Delta_prime, S, S_prime, F, J – polynomials from the same polynomial ring.

OUTPUT:

Zero if S is the first polynomial, S_prime the second polynomial, and the remaining input are the invariants and covariants of a ternary biquadratic.

EXAMPLES:

sage: R.<x,y,z> = QQ[]
sage: monomials = [x^2, x*y, y^2, x*z, y*z, z^2]
sage: def q_rnd():  return sum(randint(-1000,1000)*m for m in monomials)
sage: biquadratic = invariant_theory.ternary_biquadratic(q_rnd(), q_rnd(), [x,y,z])
sage: Delta = biquadratic.Delta_invariant()
sage: Theta = biquadratic.Theta_invariant()
sage: Theta_prime = biquadratic.Theta_prime_invariant()
sage: Delta_prime = biquadratic.Delta_prime_invariant()
sage: S = biquadratic.first().polynomial()
sage: S_prime  = biquadratic.second().polynomial()
sage: F = biquadratic.F_covariant()
sage: J = biquadratic.J_covariant()
sage: biquadratic.syzygy(Delta, Theta, Theta_prime, Delta_prime, S, S_prime, F, J)
0

If the arguments are not the invariants and covariants then the output is some (generically non-zero) polynomial:

sage: biquadratic.syzygy(1, 1, 1, 1, 1, 1, 1, x)
1/64*x^2 + 1

sage.rings.invariants.invariant_theory.transvectant(f, g, h=1, scale='default')#

Return the h-th transvectant of f and g.

INPUT:

f,g – two homogeneous binary forms in the same polynomial ring.
h – the order of the transvectant. If it is not specified, the first transvectant is returned.
scale – the scaling factor applied to the result. Possible values are 'default' and 'none'. The 'default' scaling factor is the one that appears in the output statement below, if the scaling factor is 'none' the quotient of factorials is left out.

OUTPUT:

The h-th transvectant of the listed forms \(f\) and \(g\):

\[(f,g)_h = \frac{(d_f-h)! \cdot (d_g-h)!}{d_f! \cdot d_g!}\left( \left(\frac{\partial}{\partial x}\frac{\partial}{\partial z'} - \frac{\partial}{\partial x'}\frac{\partial}{\partial z} \right)^h \left(f(x,z) \cdot g(x',z')\right) \right)_{(x',z')=(x,z)}\]

EXAMPLES:

sage: from sage.rings.invariants.invariant_theory import AlgebraicForm, transvectant
sage: R.<x,y> = QQ[]
sage: f = AlgebraicForm(2, 5, x^5 + 5*x^4*y + 5*x*y^4 + y^5)
sage: transvectant(f, f, 4)
Binary quadratic given by 2*x^2 - 4*x*y + 2*y^2
sage: transvectant(f, f, 8)
Binary form of degree -6 given by 0

The default scaling will yield an error for fields of positive characteristic below \(d_f!\) or \(d_g!\) as the denominator of the scaling factor will not be invertible in that case. The scale argument 'none' can be used to compute the transvectant in this case:

sage: R.<a0,a1,a2,a3,a4,a5,x0,x1> = GF(5)[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: f = AlgebraicForm(2, 5, p, x0, x1)
sage: transvectant(f, f, 4)
Traceback (most recent call last):
...
ZeroDivisionError
sage: transvectant(f, f, 4, scale='none')
Binary quadratic given by -a3^2*x0^2 + a2*a4*x0^2 + a2*a3*x0*x1
- a1*a4*x0*x1 - a2^2*x1^2 + a1*a3*x1^2

The additional factors that appear when scale='none' is used can be seen if we consider the same transvectant over the rationals and compare it to the scaled version:

sage: R.<a0,a1,a2,a3,a4,a5,x0,x1> = QQ[]
sage: p = a0*x1^5 + a1*x1^4*x0 + a2*x1^3*x0^2 + a3*x1^2*x0^3 + a4*x1*x0^4 + a5*x0^5
sage: f = AlgebraicForm(2, 5, p, x0, x1)
sage: transvectant(f, f, 4)
Binary quadratic given by 3/50*a3^2*x0^2 - 4/25*a2*a4*x0^2
+ 2/5*a1*a5*x0^2 + 1/25*a2*a3*x0*x1 - 6/25*a1*a4*x0*x1 + 2*a0*a5*x0*x1
+ 3/50*a2^2*x1^2 - 4/25*a1*a3*x1^2 + 2/5*a0*a4*x1^2
sage: transvectant(f, f, 4, scale='none')
Binary quadratic given by 864*a3^2*x0^2 - 2304*a2*a4*x0^2
+ 5760*a1*a5*x0^2 + 576*a2*a3*x0*x1 - 3456*a1*a4*x0*x1
+ 28800*a0*a5*x0*x1 + 864*a2^2*x1^2 - 2304*a1*a3*x1^2 + 5760*a0*a4*x1^2

If the forms are given as inhomogeneous polynomials, the homogenisation might fail if the polynomial ring has multiple variables. You can circumvent this by making sure the base ring of the polynomial has only one variable:

sage: R.<x,y> = QQ[]
sage: quintic = invariant_theory.binary_quintic(x^5+x^3+2*x^2+y^5, x)
sage: transvectant(quintic, quintic, 2)
Traceback (most recent call last):
...
ValueError: polynomial is not homogeneous
sage: R.<y> = QQ[]
sage: S.<x> = R[]
sage: quintic = invariant_theory.binary_quintic(x^5+x^3+2*x^2+y^5, x)
sage: transvectant(quintic, quintic, 2)
Binary sextic given by 1/5*x^6 + 6/5*x^5*h - 3/25*x^4*h^2
+ (50*y^5 - 8)/25*x^3*h^3 - 12/25*x^2*h^4 + (3*y^5)/5*x*h^5
+ (2*y^5)/5*h^6