An upper bound for the multiplicity and Wilf's conjecture for one-dimensional Cohen-Macaulay rings
Marco D'Anna, Alessio Moscariello
TL;DR
The paper derives a sharp upper bound for the multiplicity of a one-dimensional Cohen–Macaulay local ring with finite conductor, showing $e(R) \le (\nu-1)\ell(R/\mathfrak{c})+1$ and characterizing equality (DVR or $\ell(R/\mathfrak{c})=1$). It connects this bound to Wilf's conjecture for numerical semigroup rings via Lech's inequality and develops a Wilf-type inequality for $e(\mathfrak{c})$ in almost Gorenstein rings: $e(\mathfrak{c}) \le \nu\,\ell(R/\mathfrak{c})$. The results are interpreted through conductor and reduction frameworks, with a detailed proof strategy using canonical modules and length computations, and extended considerations to canonical reductions and higher-dimensional contexts. Overall, the work links multiplicity bounds, conductor-length data, and Wilf-type phenomena in both ring-theoretic and semigroup settings, highlighting conditions for sharpness and potential generalizations.
Abstract
In this work we provide an upper bound for the multiplicity of a one-dimensional Cohen-Macaulay ring (under certain conditions), describe the rings attaining the equality for this bound, and outline a connection with Wilf's conjecture for numerical semigroup rings. Then we prove the analogue of Wilf's conjecture for almost Gorenstein rings.