Maps between Rings

Magma includes functions for working with maps between multivariate polynomial rings. Let R=K₁[x₁, ..., x_n] and S=K₂[y₁, ..., y_m] be a polynomial rings over the fields K₁, K₂, and f : R to S a ring homomorphism.

PolyMapKernel(f) : Map -> RngMPol

Return the kernel of the map f as an ideal in the domain R, i.e., the set { a ∈R | f(a) = 0 }. This is basically the computation of the relation ideal for the polynomials defining the map and is as described in RelationIdeal.

IsInImage(f, p) : Map, RngMPolElt -> [ BoolElt ]

Given a polynomial p in S, return whether p is in the image of the map f. The algorithm is the one described on p. 82 of [AL94].

IsSurjective(f) : Map -> [ BoolElt ]

Return whether the map f is surjective. Uses the function above to check whether each codomain variable lies in the image.

Extension(phi, I): Map, RngMPol -> RngMPol

The extension of the ideal I by φ, where φ is a homomorphism from the generic of I. That is, the ideal generated by the image of I under φ.

Implicitization(phi) : Map -> RngMPol

Suppose the polynomial map φ: Kⁿ to K^m is a parametrization of a variety V, i.e., V is the image of φ in K^m. This function constructs the ideal of S corresponding to V.

The map φ maps (z₁, ..., z_n) |-> (f₁(z₁), ..., f_m(z_m)) where the z_i are the coordinates of Kⁿ. Let f: S to R be the map of polynomial rings defined by (y₁, ..., y_m) |-> (f₁(y₁), ..., f_m(y_m)). Then Implicitization(f) is the ideal of S corresponding to V.

If V is not a true variety, the function returns the smallest variety containing V (the Zariski closure of V).

The algorithm used is given on p. 97 of [CLO96]

> R<x, y> := PolynomialRing(Rationals(), 2);
> S<r, u, v, w> := PolynomialRing(Rationals(), 4);
> f := hom<S -> R |x^4, x^3*y, x*y^3, y^4>;
> Implicitization(f);
Ideal of Polynomial ring of rank 4 over Rational Field
Lexicographical Order
Variables: r, u, v, w
Basis:
[
    -r^2*v + u^3,
    r*v^2 - u^2*w,
    -u*w^2 + v^3,
    -r*w + u*v
]