2024/03/14 by Vittoria Bonanzinga, Bonanzinga, Vittoria, Shalom Eliahou +1 · 1 citation
Mathematics · Computer Science · #Commutative Algebra and Its Applications #Polynomial and algebraic computation #Algebraic Geometry and Number Theory
paper · pdf · doi:10.48550/arxiv.2403.09497
Let Rn=K[x1,…,xn] be the n-variable polynomial ring over a field K. Let Sn denote the set of monomials in Rn. A monomial u ∈ Sn is a Gotzmann monomial if the Borel-stable monomial ideal ⟨ u ⟩ it generates in Rn is a Gotzmann ideal. A longstanding open problem is to determine all Gotzmann monomials in Rn. Given u0 ∈ Sn-1, its Gotzmann threshold is the unique nonnegative integer t0=τn(u0) such that u0xnt is a Gotzmann monomial in Rn if and only if t ≥ t0. Currently, the function τn is exactly known for n ≤ 4 only. We present here an efficient procedure to determine τn(u0) for all n and all u0 ∈ Sn-1. As an application, in the critical case u0=x2d, we determine τ5(x2d) for all d and we conjecture that for n ≥ 6, τn(x2d) is a polynomial in d of degree 2n-2 and dominant term equal to that of the (n-2)-iterated binomial coefficient \binom \binom \binom d22\stackrel⋯2.