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### Multivariate Polynomials over Rational Function Fields

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.<u,v> = PolynomialRing(QQ, 'u', 'v') R.<x,y> = 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.

 2 fix typesetting William Stein 2154 ●11 ●32 ●63 http://wstein.org/

### Multivariate Polynomials over Rational Function Fields

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} y^2}


where the polynomials belong to the ring Q(u,v)[x,y].

I tried the following

B.<u,v> = PolynomialRing(QQ, 'u', 'v')
R.<x,y> = PolynomialRing(B, 'x', 'y')
I = R.ideal(v * x^2 + y, u* x * y + y^2)
g = I.groebner_basis()I.groebner_basis()


This fails with the error

TypeError: Can only reduce polynomials over fields.fields.

 3 retagged Kelvin Li 503 ●5 ●12 ●17

### Multivariate Polynomials over Rational Function Fields

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.<u,v> = PolynomialRing(QQ, 'u', 'v')
R.<x,y> = 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.