On the monotonicity of the Hilbert functions for 4-generated pseudo-symmetric monomial curves

TL;DR

This paper proves that the Hilbert function of the local ring for 4-generated pseudo-symmetric numerical semigroups is always non-decreasing, providing explicit computations and characterizations related to the tangent cone's properties.

Contribution

It offers a complete characterization of standard bases for these semigroups and confirms the non-decreasing nature of the Hilbert function even when the tangent cone is not Cohen-Macaulay.

Findings

Hilbert function is non-decreasing for the studied semigroups

Standard bases depend on parameters s_j

Explicit Hilbert function computations support the proof

Abstract

In this article we solve the conjecture "Hilbert function of the local ring for a 4 generated pseudo-symmetric numerical semigroup $\langle n_1,n_2,n_3,n_4 \rangle$ is always non-decreasing when $ n_1 < n_2 < n_3 < n_4$". We give a complete characterization to the standard bases when the tangent cone is not Cohen-Macaulay by showing that the number of elements in the standard basis depends on some parameters $s_j$ 's we define. Since the tangent cone is not Cohen-Macaulay, non-decreasingness of the Hilbert fuction was not guaranteed, we proved the non-decreasingness from our explicit Hilbert Function computation.