On Apr 5, 2:10 pm, "Johan S. R. Nielsen" <santaph...@gmail.com> wrote:
> Oops, continuing:
>
> more precisely, we wish to find a q in Q[Y1, Y2] such that q(f1, f2) =
> g. In this case, we have
> q(Y1, Y2) = Y1^2 + Y1*Y2 - Y2
> as a solution, as
> f1^2 + f1*f2 - f2 = g
This is an elimination problem. Note that it is not enough that g
belongs to the ideal to be able to write it in the desired form. You
instead want to check if g belongs to the ring Q[f1,f2] that is a
different problem.
I can think of the following:
First you add new variables for your polynomials f1,f2 and g, call
them y1,y2,z and a polynomial ring with a block elimination term
order.
sage: K=PolynomialRing(QQ, 'x,z,y1,y2',order=TermOrder('degrevlex',
2)+TermOrder('degrevlex',2))
sage: K.inject_variables()
Defining x, z, y1, y2
In this ring, x and z are greater than y1,y2 now construct the ideal
defining your polynomials
I=Ideal(x^2+1-y1, x+3-y2, x^4+x^3+4*x^2+x+3-z)
If we eliminate x from this ideal we will get the ideal of algebraic
dependence on f1,f2,g
sage: J=I.elimination_ideal([x])
sage: J
Ideal (y2^2 - y1 - 6*y2 + 10, z - y1^2 - y1*y2 + y1) of Multivariate
Polynomial Ring in x, z, y1, y2 over Rational Field
If I am not making any mistake, the reduction of z under this ideal
with this term ordering should give the desired polynomial.
sage: J.reduce(z)
y1^2 + y1*y2 - 4*y1 + 3
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