Eisenbud–Green–Harris and lex-plus-powers

The explanation

Take polynomials in two letters, x and y. Set x² = 0 and y² = 0, and count what survives in each degree. In degree 0 there is 1. In degree 1 there are x and y, so 2. In degree 2 only xy survives, so 1. In degree 3 nothing survives. This list, 1, 2, 1, is called the Hilbert function. The ideal here, written (x², y²), is the set of all multiples of x² and y².

This ideal contains x² and y³, a regular sequence of degrees 2 and 3. Regular means each one is a genuinely new condition on top of the ones before. The lex-plus-powers recipe now builds a model ideal from monomials only. Start with the pure powers x² and y³. Then, degree by degree, add the earliest monomials in dictionary order until the counts match. In degree 2 that adds xy. The model is (x², xy, y³), and it also leaves 1, 2, 1.

Now compare relations. The original ideal has two generators of degree 2 and one relation between them, of degree 4. The model has generators of degrees 2, 2 and 3, and relations of degrees 3 and 4. Counted degree by degree, the model never has fewer. These counts are called graded Betti numbers.

The lex-plus-powers conjecture predicts one extremal ideal for both kinds of dataSource 1. Its Hilbert-function half is the Eisenbud–Green–Harris conjectureSource 2. Eisenbud, Green and Harris raised it in work on higher Castelnuovo theorySource 3. The Betti version is generally credited to Charalambous and EvansSource 4. With no regular sequence at all, the matching statement for lex ideals was already a classical theoremSource 5.

Many cases were known. Mermin and Murai proved the comparison when the regular sequence consists of monomialsSource 6. Other results needed the degrees to grow quickly, each roughly at least the sum of the earlier ones. This work imposes no such growth conditionSource 7.

The companion paper claims the Artinian Eisenbud–Green–Harris conjecture over the complex numbersSource 8. Artinian means the regular sequence is as long as the number of letters, so only finitely much survives. The main paper claims the full Artinian lex-plus-powers conjecture over the complex numbersSource 9. It then removes the Artinian condition and extends to every field of characteristic zero, using known reductionsSource 10. Characteristic zero means 1 + 1 + … + 1 never equals 0.

The one obstacle is that matching the counts says nothing yet about the relationsSource 11. The trick is to swap the letters for new commuting linear expressions. Their coefficients come from a division algebra, a number system where every nonzero element can be divided bySource 12. It also keeps an ordered list of building blocks for all polynomials. A resolution of the original ideal then carries over cleanly, with every Betti number multiplied by the same factorSource 13. That produces a monomial ideal with matching counts and no fewer relations. The final monomial comparison was already known, and the paper says soSource 14.

How sure is this? No Lean formalization is listed for either paper. The claims rest on the papers' own arguments and the earlier reductions they cite. No specialist review of these papers has been published.

What it doesn't show. This does not cover fields of positive characteristic, such as arithmetic modulo a prime.