Polynomials#
In this section we illustrate how to create and use polynomials in Sage.
Univariate Polynomials#
There are three ways to create polynomial rings.
sage: R = PolynomialRing(QQ, 't')
sage: R
Univariate Polynomial Ring in t over Rational Field
This creates a polynomial ring and tells Sage to use (the string)
‘t’ as the indeterminate when printing to the screen. However, this
does not define the symbol t for use in Sage, so you cannot use it to
enter a polynomial (such as \(t^2+1\)) belonging to R.
An alternate way is
sage: S = QQ['t']
sage: S == R
True
This has the same issue regarding t.
A third very convenient way is
sage: R.<t> = PolynomialRing(QQ)
or
sage: R.<t> = QQ['t']
or even
sage: R.<t> = QQ[]
This has the additional side effect that it defines the variable
t to be the indeterminate of the polynomial ring, so you can
easily construct elements of R, as follows. (Note that the third
way is very similar to the constructor notation in Magma, and just
as in Magma it can be used for a wide range of objects.)
sage: poly = (t+1) * (t+2); poly
t^2 + 3*t + 2
sage: poly in R
True
Whatever method you use to define a polynomial ring, you can recover the indeterminate as the \(0^{th}\) generator:
sage: R = PolynomialRing(QQ, 't')
sage: t = R.0
sage: t in R
True
Note that a similar construction works with the complex numbers:
the complex numbers can be viewed as being generated over the real
numbers by the symbol i; thus we have the following:
sage: CC
Complex Field with 53 bits of precision
sage: CC.0 # 0th generator of CC
1.00000000000000*I
For polynomial rings, you can obtain both the ring and its generator, or just the generator, during ring creation as follows:
sage: R, t = QQ['t'].objgen()
sage: t = QQ['t'].gen()
sage: R, t = objgen(QQ['t'])
sage: t = gen(QQ['t'])
Finally we do some arithmetic in \(\QQ[t]\).
sage: R, t = QQ['t'].objgen()
sage: f = 2*t^7 + 3*t^2 - 15/19
sage: f^2
4*t^14 + 12*t^9 - 60/19*t^7 + 9*t^4 - 90/19*t^2 + 225/361
sage: cyclo = R.cyclotomic_polynomial(7); cyclo
t^6 + t^5 + t^4 + t^3 + t^2 + t + 1
sage: g = 7 * cyclo * t^5 * (t^5 + 10*t + 2)
sage: g
7*t^16 + 7*t^15 + 7*t^14 + 7*t^13 + 77*t^12 + 91*t^11 + 91*t^10 + 84*t^9
+ 84*t^8 + 84*t^7 + 84*t^6 + 14*t^5
sage: F = factor(g); F
(7) * t^5 * (t^5 + 10*t + 2) * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)
sage: F.unit()
7
sage: list(F)
[(t, 5), (t^5 + 10*t + 2, 1), (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1, 1)]
Notice that the factorization correctly takes into account and records the unit part.
If you were to use, e.g., the R.cyclotomic_polynomial function a
lot for some research project, in addition to citing Sage you should
make an attempt to find out what component of Sage is being used to
actually compute the cyclotomic polynomial and cite that as well.
In this case, if you type R.cyclotomic_polynomial?? to see the
source code, you’ll quickly see a line f = pari.polcyclo(n) which
means that PARI is being used for computation of the cyclotomic
polynomial. Cite PARI in your work as well.
Dividing two polynomials constructs an element of the fraction field (which Sage creates automatically).
sage: x = QQ['x'].0
sage: f = x^3 + 1; g = x^2 - 17
sage: h = f/g; h
(x^3 + 1)/(x^2 - 17)
sage: h.parent()
Fraction Field of Univariate Polynomial Ring in x over Rational Field
Using Laurent series, one can compute series expansions in the
fraction field of QQ[x]:
sage: R.<x> = LaurentSeriesRing(QQ); R
Laurent Series Ring in x over Rational Field
sage: 1/(1-x) + O(x^10)
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + O(x^10)
If we name the variable differently, we obtain a different univariate polynomial ring.
sage: R.<x> = PolynomialRing(QQ)
sage: S.<y> = PolynomialRing(QQ)
sage: x == y
False
sage: R == S
False
sage: R(y)
x
sage: R(y^2 - 17)
x^2 - 17
The ring is determined by the variable. Note that making another
ring with variable called x does not return a different ring.
sage: R = PolynomialRing(QQ, "x")
sage: T = PolynomialRing(QQ, "x")
sage: R == T
True
sage: R is T
True
sage: R.0 == T.0
True
Sage also has support for power series and Laurent series rings over any base ring. In the following example, we create an element of \(\GF{7}[[T]]\) and divide to create an element of \(\GF{7}((T))\).
sage: R.<T> = PowerSeriesRing(GF(7)); R
Power Series Ring in T over Finite Field of size 7
sage: f = T + 3*T^2 + T^3 + O(T^4)
sage: f^3
T^3 + 2*T^4 + 2*T^5 + O(T^6)
sage: 1/f
T^-1 + 4 + T + O(T^2)
sage: parent(1/f)
Laurent Series Ring in T over Finite Field of size 7
You can also create power series rings using a double-brackets shorthand:
sage: GF(7)[['T']]
Power Series Ring in T over Finite Field of size 7
Multivariate Polynomials#
To work with polynomials of several variables, we declare the polynomial ring and variables first.
sage: R = PolynomialRing(GF(5),3,"z") # here, 3 = number of variables
sage: R
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5
Just as for defining univariate polynomial rings, there are alternative ways:
sage: GF(5)['z0, z1, z2']
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5
sage: R.<z0,z1,z2> = GF(5)[]; R
Multivariate Polynomial Ring in z0, z1, z2 over Finite Field of size 5
Also, if you want the variable names to be single letters then you can use the following shorthand:
sage: PolynomialRing(GF(5), 3, 'xyz')
Multivariate Polynomial Ring in x, y, z over Finite Field of size 5
Next let’s do some arithmetic.
sage: z = GF(5)['z0, z1, z2'].gens()
sage: z
(z0, z1, z2)
sage: (z[0]+z[1]+z[2])^2
z0^2 + 2*z0*z1 + z1^2 + 2*z0*z2 + 2*z1*z2 + z2^2
You can also use more mathematical notation to construct a polynomial ring.
sage: R = GF(5)['x,y,z']
sage: x,y,z = R.gens()
sage: QQ['x']
Univariate Polynomial Ring in x over Rational Field
sage: QQ['x,y'].gens()
(x, y)
sage: QQ['x'].objgens()
(Univariate Polynomial Ring in x over Rational Field, (x,))
Multivariate polynomials are implemented in Sage using Python dictionaries and the “distributive representation” of a polynomial. Sage makes some use of Singular [Si], e.g., for computation of gcd’s and Gröbner basis of ideals.
sage: R, (x, y) = PolynomialRing(RationalField(), 2, 'xy').objgens()
sage: f = (x^3 + 2*y^2*x)^2
sage: g = x^2*y^2
sage: f.gcd(g)
x^2
Next we create the ideal \((f,g)\) generated by \(f\)
and \(g\), by simply multiplying (f,g) by R (we could
also write ideal([f,g]) or ideal(f,g)).
sage: I = (f, g)*R; I
Ideal (x^6 + 4*x^4*y^2 + 4*x^2*y^4, x^2*y^2) of Multivariate Polynomial
Ring in x, y over Rational Field
sage: B = I.groebner_basis(); B
[x^6, x^2*y^2]
sage: x^2 in I
False
Incidentally, the Gröbner basis above is not a list but an immutable sequence. This means that it has a universe, parent, and cannot be changed (which is good because changing the basis would break other routines that use the Gröbner basis).
sage: B.universe()
Multivariate Polynomial Ring in x, y over Rational Field
sage: B[1] = x
Traceback (most recent call last):
...
ValueError: object is immutable; please change a copy instead.
Some (read: not as much as we would like) commutative algebra is available in Sage, implemented via Singular. For example, we can compute the primary decomposition and associated primes of \(I\):
sage: I.primary_decomposition()
[Ideal (x^2) of Multivariate Polynomial Ring in x, y over Rational Field,
Ideal (y^2, x^6) of Multivariate Polynomial Ring in x, y over Rational Field]
sage: I.associated_primes()
[Ideal (x) of Multivariate Polynomial Ring in x, y over Rational Field,
Ideal (y, x) of Multivariate Polynomial Ring in x, y over Rational Field]