Hilbert Series for Ideals Generated by Generic Forms

A natural way to use computer calculations in mathematics is to solve lots of special cases of a problem, trying to see a pattern, and then find a theorem which one can guess from the results. In some "generic" cases, however, it is not hard to see that a statement is true on an open subset of some algebraic set, but the real difficulty is to prove that this open set is not empty. Thus, to prove a theorem, one only needs one example. We present an application to this technique, where we by use of computers are able to prove mathematical theorems. The question we study is the following: Suppose that we are given some positive integers n ,g,d1,…,dg. How large can an ideal generated by forms f1 ,…,fg of degrees d1 ,…,dg in the polynomial ring in n variables over be? There is a conjecture, and some partial results are known. We prove the conjecture in some more cases, and investigate if powers of linear forms are sufficiently general to be generators for the largest possible ideals. Our method depends heavily on computer calculations.