[GAP Forum] Polynomial ring of indeterminates

[Alexander Hulpke]

Dear Forum, Dear Gabor,

> On Apr 20, 2015, at 4/20/15 10:04, Nagy Gabor <nagyg at math.u-szeged.hu> wrote:
> 
> Dear Forum,
> 
> Assume that the indetermines x,y are defined using the command <PolynomialRing> as follows:
> 
> q:=7;
> R:=PolynomialRing(GF(q^2),["x","y"]);
> x:=R.1; y:=R.2;
> 
> Is it then possible to get back the base ring R from x or y?


Alas this is not possible. The reason for this is that it internally implements a polynomial ring for a particular elements family (i.e. in this case it would be the polynomial ring for the algebraic of a closed field in characteristics 7) in countably many indeterminants.

Doing so allows us to perform arithmetic amongst all these polynomials without conversion functions (for example if you introduce a new variable later), however it means that polynomials do not have anything which points back to the ring from whose generators they were created.

What you can do however is to get the index number of the variables of a bivariate polynomial F from this polynomial and that way re-create the indeterminates. So your program would be:

rho:=function(f)
local indets,fam;
  indets:=OccuringVariableIndices(f);
  if Length(indets)<2 then Error("polynomial is not bivariate");fi;
  fam:=CoefficientsFamily(FamilyObj(f));
  indets:=List(indets,x->UnivariateLaurentPolynomialByCoefficients(
    fam,[One(fam)],1,x));
  return Resultant(f,indets[1]^2-indets[2]^3+indets[2],indets[2]);
end;

Regards,

   Alexander

— Alexander Hulpke, Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
http://www.math.colostate.edu/~hulpke