Algorithm for Polynomial Reduction in a Quotient Ring

Question

Any reference or suggestion for the following problem would be greatly appreciated. I am working on the quotient ring $Q=R[X_1,\dots,X_n]/<f_1,\dots,f_k>$. Given polynomials $p$ and $q$ I want to obtain a polynomial $u$ such that

$$ q\cdot u \equiv p \text{ in } Q .$$

In particular, I have a quotient of polynomials $\frac{p}{q}$ and I would like write it as a simple polynomial not involving any fraction. This should be possible by adding a term that belongs to the ideal and is therefore $\equiv0$.

Answer 1

This is a Gröbner basis problem. You are trying to find an expression for $p$ that it is contained in the ideal generated by $f_1,\dots,f_k,q$. Assuming $u$ exists, then it can be taken to be the coefficient of $q$ in such an expression. One can do this in practice using the // command in Macaulay2:

http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.6/share/doc/Macaulay2/Macaulay2Doc/html/___Matrix_sp_sl_sl_sp__Matrix.html

This isn't necessarily unique unless you have further properties, like say $Q$ is an integral domain.