Graded Betti Numbers of the Logarithmic Derivation Module

TL;DR

This paper investigates the graded Betti numbers of the logarithmic derivation module associated with a homogeneous polynomial, extending classical formulas from free cases to non-free and quasi-homogeneous cases, and exploring potential generalizations.

Contribution

It generalizes the classical sum formula for exponents of free modules to non-free and quasi-homogeneous cases, providing new insights into the structure of the derivation module.

Findings

Generalized the sum formula for exponents to non-free modules

Extended the formula to quasi-homogeneous polynomials

Explored potential generalizations for arbitrary polynomials

Abstract

Let $Q\in \K[x_1,...,x_n] = S$ be a homogeneous polynomial of degree $d$. The freeness of the logarithmic derivation module, $D(Q)$, and of its natural generalizations, has been widely studied. In the free case, $D(Q) \simeq \bigoplus_{i=1}^n S(-d_i)$ where the $d_i$'s are the exponents of the module; and as a direct consequence of the Saito-Ziegler criterion, the formula $d = \sum_i d_i$ holds. In this paper we give a generalization of this formula in the non-free case. Moreover, we show that an equivalent formula is also true in the quasi-homogeneous case, and show to what extent it can be generalized for arbitrary polynomials.