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polynomial decomposition

Assume i have some polynomials P1,...,Pn and a polynomial P. Sage is able to tell whether P is in the ideal generated by P1,...,Pn. When the answer is True, i would like to obtain a certificate, that is a list of polynomials Q1,...,Qn such that P = P1*Q1 + ... + Pn*Qn. Is there an easy way to get this ?

polynomial decomposition

Assume i have some multivariate polynomials P1,...,Pn and a polynomial P over some ring R. Sage is able to tell whether P is in the ideal generated by P1,...,Pn by writing P in R.ideal([P1,...,Pn]). When the answer is True, i would like to obtain a certificate, that is a list of polynomials Q1,...,Qn such that P = P1*Q1 + ... + Pn*Qn. Is there an easy way to get this ?? If possible i would like the Qi to have small coefficients.

polynomial decomposition

Assume i have some multivariate polynomials P1,...,Pn and a polynomial P over some ring R. Sage is able to tell whether P is in the ideal generated by P1,...,Pn by writing P in R.ideal([P1,...,Pn]), where R is the polynomial ring containing the Pi. When the answer is True, i would like to obtain a certificate, that is a list of polynomials Q1,...,Qn such that P = P1*Q1 + ... + Pn*Qn. Is there an easy way to get this ? If possible i would like the Qi to have small coefficients.