Eisenbud-Green-Harris conjecture (lex-plus-powers)

### What is the conjecture

Let $S = k[x_1, \dots, x_n]$ with $x_1 > x_2 > \dots > x_n$, and let $I \subseteq S$ be a
homogeneous ideal. Write $\mathrm{HF}(I; j) = \dim_k I_j$ for its Hilbert function. A monomial
ideal $L$ is *lex* if for monomials $u, v$ of the same degree with $u >_{\mathrm{lex}} v$ we have
$u \in L$ whenever $v \in L$. Macaulay's theorem says every homogeneous ideal has the same
Hilbert function as a unique lex ideal.

**Conjecture (Eisenbud-Green-Harris).** If $I$ contains a regular sequence of forms of degrees
$d_1 \le \dots \le d_c$, then there is a lex ideal $L$ such that $I$ has the same Hilbert function
as the *lex-plus-powers* ideal $L + (x_1^{d_1}, \dots, x_c^{d_c})$.

Eisenbud, Green and Harris proposed this to generalise a theorem of Castelnuovo and
Cayley-Bacharach type theorems for zero-dimensional subschemes of $\mathbb P^m$. It refines
Macaulay's theorem, which is the case $c = 0$, by using the extra information that $I$ contains a
complete intersection of known degrees. A typical consequence: if $\Gamma \subseteq \mathbb P^m$
is zero-dimensional and cut out by $m + 1$ quadrics, then $\deg(\Gamma) \le 2^m - 2^{m-2}$.

The conjecture is open. Known cases include: the regular sequence is monomial, equivalently $I$
already contains $(x_1^{d_1}, \dots, x_c^{d_c})$ (Clements-Lindström); the degrees satisfy
$d_i \ge \sum_{j

Start by reviewing MvPolynomial.homogeneousSubmodule, Ideal.IsHomogeneous, MonomialOrder.toLex, and RingTheory.Sequence.IsRegular. Add the missing Hilbert-function, monomial-ideal, and lex-ideal definitions, then formalize the Eisenbud-Green-Harris statement and verify it builds with the repository's Lean checks.

Written by the indexing model from the issue text.