On the Lex-plus-powers Conjecture

Abstract

Let $S$ be a polynomial ring over a field and $I\subseteq S$ a homogeneous ideal containing a regular sequence of forms of degrees $d_1, \ldots, d_c$. In this paper we prove the Lex-plus-powers Conjecture when the field has characteristic 0 for all regular sequences such that $d_i \geq \sum_{j=1}^{i-1} (d_j-1)+1$ for each $i$; that is, we show that the Betti table of $I$ is bounded above by the Betti table of the lex-plus-powers ideal of $I$. As an application, when the characteristic is 0, we obtain bounds for the Betti numbers of any homogeneous ideal containing a regular sequence of known degrees, which are sharper than the previously known ones from the Bigatti-Hulett-Pardue Theorem.