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Maps between Polynomial Rings


Maps between Polynomial Rings






Magma includes functions for working with maps between multivariate
polynomial rings.
Let R=K_1[x_1, ..., x_n] and S=K_2[y_1, ..., 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 f as an ideal in R,
i.e., the set { a in R | f(a) = 0 }.



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



Given an element p in S, returns true if p is in the image of f.



IsSurjective(f) : Map  -> [ BoolElt ]



Return true if and only if f is surjective.



Implicitization(f) : 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, ..., z_n) |-> (f_1(z_1), ..., 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, ..., y_m) |-> (f_1(y_1), ..., 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).





Example RngMPol_Map1 (H29E29)

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) |-> (x^4, x^3y, xy^3, y^4).  
This example is taken from Adams and Loustanau, "An Introduction to
Gröbner Bases", p. 92.





> 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
]





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