# Wilf's conjecture and Macaulay's theorem

## TL;DR

This paper proves Wilf's conjecture for numerical semigroups with conductor at most three times the multiplicity, using Macaulay's theorem, and discusses its asymptotic validity as the genus increases.

## Contribution

It establishes Wilf's conjecture for a new class of semigroups and links it to Macaulay's theorem, advancing understanding of the conjecture's scope.

## Key findings

- Wilf's conjecture holds if c ≤ 3m.
- The conjecture is asymptotically true as genus g(S) increases.
- Uses Macaulay's theorem to analyze semigroup properties.

## Abstract

Let S $\subseteq$ N be a numerical semigroup with multiplicity m = min(S \ {0}), conductor c = max(N \ S) + 1 and minimally generated by e elements. Let L be the set of elements of S which are smaller than c. Wilf conjectured in 1978 that |L| is bounded below by c/e. We show here that if c $\le$ 3m, then S satisfies Wilf's conjecture. Combined with a recent result of Zhai, this implies that the conjecture is asymptotically true as the genus g(S) = |N \ S| goes to infinity. One main tool in this paper is a classical theorem of Macaulay on the growth of Hilbert functions of standard graded algebras.