Eisenbud–Green–Harris lex-plus-powers conjecture

Prove that every homogeneous ideal I in a standard graded polynomial ring containing a regular sequence of degrees a_1≤\cdots≤a_m has the same Hilbert function as some lex-plus-powers ideal (x_1^{a_1},\ldots,x_m^{a_m})+J, where J is a lex ideal.

Background

The Eisenbud–Green–Harris conjecture is used as an algebraic mechanism for replacing an arbitrary ideal containing a regular sequence by a lex-plus-powers model with the same Hilbert function. In the paper, this conjecture is invoked for degree sequence (d^{m−k},q^k).

The paper establishes that the relevant EGH statement implies the affine algebraic conjecture, but the general Eisenbud–Green–Harris conjecture remains unresolved; only several special cases are listed as known.