# How to implement the multivariable division algorithm without passing to Grobner bases?

Hi, I'm new to Sage, and I'm wondering how to implement the multivariable division algorithm in Sage. I pulled up the "Multivariate Polynomials via libSINGULAR" page of the Sage Reference Manual v7.1, but it wasn't helpful. What I'm wanting is a generalization of the quo_rem command that can take in more than one argument on the right and follows the division algorithm with respect to a fixed monomial ordering and the order that the polynomials are entered in. Is there any set of commands that does that for me? If so, would you please include the code, say for the following example?

Divide the polynomial y

x^2 + xy^2 + y^2 by xy-1 and y2 -1 (in that order) using the lexicographic ordering with x>y. I would like to process more complicated examples, perhaps with that order and dividing by 8 things at once rather than 2.

I've learned about the p.mod(I) and p.reduce(I) commands where p is a polynomial and I is an ideal. The problem with those is that they seem to pass to a Grobner basis for I to get a "canonical" remainder rather than the remainder we'd get from the given order of the polynomials, as I tested switching the order of the polynomials in defining an ideal I and it did not change my answer for p.mod(I) or p.reduce(I).

Thanks!