On the Gotzmann threshold of monomials

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 its associated Borel-stable monomial ideal 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 non-negative integer t0 = τn(u0) such that u0xt n 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 = xd 2 , we determine τ5(xd 2 ) for all d and we conjecture that for n ≥ 6, τn(xd 2 ) is a polynomial in d of degree 2n−2 and dominant term equal to that of the (n − 2)-iterated binomial coefficient ( d 2) 2 ··· 2