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

Abstract

In this article we solve the following problem: “The Hilbert function of thelocal ring of a 4-generated pseudo-symmetric numerical semigroup is alwaysnon-decreasing.” We give a complete characterization of the standard baseswhen the tangent cone is not Cohen-Macaulay by showing that the number ofelements in the standard basis depends on some parameters we define. Sincethe tangent cone is not Cohen-Macaulay, non-decreasingness of the Hilbertfunction was not guaranteed, thus we proved the non-decreasingness fromour explicit Hilbert function computation.