Maps between Rings

Magma includes functions for working with maps between multivariate polynomial rings. Let \(R=K_1[x_1,\ldots,x_n]\) and \(S=K_2[y_1,\ldots, y_m]\) be a polynomial rings over the fields \(K_1\), \(K_2\), 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 \(\{ a \in R | f(a) = 0 \}.\) 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 [Adams and Loustaunau, 1994].

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 \(\phi\), where \(\phi\) is a homomorphism from the generic of \(I\). That is, the ideal generated by the image of \(I\) under \(\phi\).

Implicitization(phi): Map -> RngMPol

Suppose the polynomial map \(\phi: K^n \to K^m\) is a parametrization of a variety \(V\), i.e., \(V\) is the image of \(\phi\) in \(K^m\). This function constructs the ideal of \(S\) corresponding to \(V\). The map \(\phi\) maps \((z_1, \dots, z_n) \mapsto (f_1(z_1), \dots, f_m(z_m))\) where the \(z_i\) are the coordinates of \(K^n\). Let \(f: S \to R\) be the map of polynomial rings defined by \((y_1, \dots, y_m) \mapsto (f_1(y_1), \dots, f_m(y_m))\). 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 [Cox et al., 1996]

Example: Map1 (ex-a72e07)

We demonstrate the use of the function Implicitization for the variety defined by \(\phi: Q[x,y] \to Q[r,u,v,w]\), \((x,y) \mapsto (x^4, x^3y, xy^3, y^4)\). This example is taken from [Adams and Loustaunau, 1994, Ex. 2.5.4].

> 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
]

Run in calculatorCopy input