Optimizing Vu’s periodicity threshold in the shifted family of monomial curves

Improve the best-known universal upper bound on the least integer j such that the Betti numbers of affine monomial curves defined by the shifted sequences j + a_1 < ⋯ < j + a_n become periodic in j, as established by Vu (2014).

Background

Vu proved that the Betti numbers in the shifted family of monomial curves eventually become periodic and gave an explicit threshold N beyond which periodicity occurs. The authors improve the bound for equality of Betti numbers between projective and affine charts (a key step in Vu’s approach) and note computational evidence suggesting further optimization of the periodicity threshold.

They explicitly propose refining Vu’s bound to achieve a better least value of j guaranteeing periodic behavior of Betti numbers in the shifted family.