[GAP Forum] Polynomial ring of indeterminates

[Nagy Gabor]

Dear Alexandre, dear Stefan,

Thanks for explaining the way GAP works with indeterminates and the 
ideas behind. It seems that I will have to change my approach into a 
more GAPish one. :-)

Best regards,

Gabor


On 2015-04-20 19:42, Alexander Hulpke wrote:
> 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
>
>
>