The residuals of lex plus powers ideals and the Eisenbud–Green–Harris conjecture

Abstract

The $n$-type vectors introduced by Geramita, Harima, and Shin are in 1—1 correspondence with the Hilbert functions of Artinian lex ideals. Letting $\mathbb{A} =\{ a_1,\ldots,a_n\}$ define the degrees of a regular sequence, we construct $\mathrm{lpp}_{\le }(\mathbb{A})$-vectors which are in 1—1 correspondence with the Hilbert functions of certain lex plus powers ideals (depending on $\mathbb{A}$). This construction enables us to show that the residual of a lex plus powers ideal in an appropriate regular sequence is again a lex plus powers ideal. We then use this result to show that the Eisenbud–Green–Harris conjecture is equivalent to showing that lex plus powers ideals have the largest last graded Betti numbers (it is well known that the Eisenbud–Green–Harris conjecture is equivalent to showing that lex plus powers ideals have the largest first graded Betti numbers).