Multivariate division algorithm

Summary

The multivariate division algorithm divides a polynomial in several variables by an ordered list of polynomials, with respect to a fixed monomial order. It writes the polynomial as a combination of the divisors plus a remainder. No term of the remainder is divisible by the leading term of a divisor. It extends the division algorithm for polynomials in one variable. In general the remainder depends on the order of the divisors, but for a Gröbner basis it is unique, and it then decides ideal membership.