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

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The n-type vectors introduced by Geramita, Harima and Shin are in 1-1 correspondence with the Hilbert functions Artinian of lex ideals. Letting A ={a₁,..., aₙ} define the degrees of a regular sequence, we construct lpp-vectors which are in 1-1 correspondence with the Hilbert functions of certain lex plus powers ideals (depending on 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).