2.7.2 Multivariate Polynomials

To work with polynomials of several variables, we declare the polynomial ring and variables first, in one of two ways.

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^2*y^2, x^6]
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.parent()
Category of sequences in Multivariate Polynomial Ring in x, y over Rational 
Field
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]

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