A Divide and Conquer Method to Compute Binomial Ideals

A binomial is a polynomial with at most two terms. In this paper, we give a divide-and-conquer strategy to compute binomial ideals. This work is a generalization of the work done by the authors in [12, 13] and is motivated by the fact that any algorithm to compute binomial ideals spends a significant amount of time computing Gröbner basis and that Gröbner basis computation is very sensitive to the number of variables in the ring. The divide and conquer strategy breaks the problem into subproblems in rings of lesser number of variables than the original ring. We apply the framework on five problems – radical, saturation, cellular decomposition, minimal primes of binomial ideals, and computing a generating set of a toric ideal.