# Multivariate Polynomials over Rational Function Fields

 2 Is it possible to define multivariate polynomials where the coefficients lie in a rational function field and do Groebner basis computations on them? Maple, Reduce and Axiom support this. For example I would like to be able to compute the Groebner basis of the polynomials {v * x^2 + y, u* x * y + y^2}  where the polynomials belong to the ring Q(u,v)[x,y]. I tried the following B. = PolynomialRing(QQ, 'u', 'v') R. = PolynomialRing(B, 'x', 'y') I = R.ideal(v * x^2 + y, u* x * y + y^2) g = I.groebner_basis()  This fails with the error TypeError: Can only reduce polynomials over fields.  asked Aug 26 '10 Sameer Agarwal 21 ● 2 ● 3 Kelvin Li 443 ● 10 ● 17

 4 Make u and v be in the Fraction field: sage: B. = Frac(QQ['u,v']) sage: R. = PolynomialRing(B, 'x', 'y') sage: I = R.ideal(v * x^2 + y, u* x * y + y^2) sage: g = I.groebner_basis() sage: g [y^3 + u^2/v*y^2, x^2 + 1/v*y, x*y + 1/u*y^2]  posted Aug 26 '10 William Stein 1369 ● 5 ● 19 ● 44 http://wstein.org/ Thanks, that works perfectly. Now my quest for the Elimination ideal I2 = I.elimination_ideal([x]) fails with TypeError: Cannot call Singular function 'eliminate' with ring parameter of type '' Sameer Agarwal (Aug 26 '10)
 1 When I try this, the error I get is a little more informative: sage: I2 = I.elimination_ideal([x]) ... TypeError: Cannot call Singular function 'eliminate' with ring parameter of type ''  The function eliminte comes from sage: eliminate = sage.libs.singular.ff.eliminate  which is a wrapper for the Singular function and is supposed to convert Sage's rings to rings that Singular understands . . . however it seems that this wrapper does not understand the _polydict_domain rings. Perhaps one could convert R to a ring that the wrapper does understand (or file this as a bug in the wrapper and fix it). Checking, I see that there is a ring type MPolynomialRing_libsingular, but it only allows base rings from a very small list (like ZZ and finite fields). Maybe someone who knows more about the Singular interface can help here? posted Aug 26 '10 niles 3605 ● 7 ● 45 ● 101 http://nilesjohnson.net/

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