2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/76068The $n$-type vectors introduced by Geramita, Harima and Shin are in 1-1 correspondence with the Hilbert functions Artinian of lex ideals. Letting $\mathbb{A} =\{a_1,..., a_n\}$ define the degrees of a regular sequence, we construct ${\rm lpp}_{\le\mathbb{A}}$-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).32 pagesCommutative Algebra13F20 (Primary) 13D02, 13D40 (Secondary)text