A study of the lex plus powers conjecture

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An { a 1,…, a n }-lex plus powers ideal is a monomial ideal in I⊂ k[ x 1,…, x n ] which minimally contains the regular sequence x 1 a 1 ,…, x n a n and such that whenever m∈ R t is a minimal generator of I and m′∈ R t is greater than m in lex order, then m′∈ I. Conjectures of Eisenbud et al. and Charalambous and Evans predict that after restricting to ideals containing a regular sequence in degrees { a 1,…, a n }, then { a 1,…, a n }-lex plus powers ideals have extremal properties similar to those of the lex ideal. That is, it is proposed that a lex plus powers ideal should give maximum possible Hilbert function growth (Eisenbud et al.), and, after fixing a Hilbert function, that the Betti numbers of a lex plus powers ideal should be uniquely largest (Charalambous, Evans). The first of these assertions would extend Macaulay's theorem on Hilbert function growth, while the second improves the Bigatti, Hulett, Pardue theorem that lex ideals have largest graded Betti numbers. In this paper we explore these two conjectures. First we give several equivalent forms of each statement. For example, we demonstrate that the conjecture for Hilbert functions is equivalent to the statement that for a given Hilbert function, lex plus powers ideals have the most minimal generators in each degree. We use this result to prove that it is enough to show that lex plus powers ideals have the most minimal generators in the highest possible degree. A similar result holds for the stronger conjecture. In this paper we also prove that if the weaker conjecture holds, then lex plus powers ideals are guaranteed to have largest socles. This suffices to show that the two conjectures are equivalent in dimension ⩽3, which proves the monomial case of the conjecture for Betti numbers in those degrees. In dimension 2, we prove both conjectures outright.