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arXiv · 2302.09356

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

Nil Şahin
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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.

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Nil Şahin. 2023-02-18. On the monotonicity of the Hilbert functions for 4-generated pseudo-symmetric monomial curves. https://doi.org/10.1080/00927872.2024.2360710

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