Wilf’s conjecture gives the relationship between the embedding dimension, the Frobenius number, and the genus of a numerical semigroup. Consider a numerical semigroup S = <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\langle a_1,a_2,\dots ,a_d \rangle $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">⟨</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>a</mi> <mi>d</mi> </msub> <mo stretchy="false">⟩</mo> </mrow> </math></EquationSource> </InlineEquation> minimally generated by d coprime positive integers, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a_1 $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> denotes the multiplicity of S. In 2015, Moscariello and Sammartano [<CitationRef CitationID="CR9">9</CitationRef>] established the validity of Wilf’s inequality for all those numerical semigroups where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$a_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> is sufficiently large and all the prime divisors of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$a_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> are greater than or equal to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>, for every fixed value of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation>=<InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\big \lceil \frac{a_1}{d} \big \rceil .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">⌈</mo> </mrow> <mfrac> <msub> <mi>a</mi> <mn>1</mn> </msub> <mi>d</mi> </mfrac> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">⌉</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> In this article, we relax the arithmetic constraint on the prime divisors of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$a_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> by replacing it with the more natural condition <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\gcd (a_1,a_2) = 1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">gcd</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> thereby significantly enlarging the family of numerical semigroups known to satisfy Wilf’s Conjecture. Moreover, under this hypothesis, we sharpen the bound on <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$a_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>a</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> appearing in the work of Moscariello and Sammartano. As a consequence, our results not only extend their theorem to a wider class of numerical semigroups but also demonstrate that Wilf’s inequality holds under strictly improved numerical bounds.

A note on Wilf’s Conjecture

期刊论文
发表日期：

2026年5月22日

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Tapas Chatterjee,
Palak Narula

摘要

Wilf’s conjecture gives the relationship between the embedding dimension, the Frobenius number, and the genus of a numerical semigroup. Consider a numerical semigroup S = $\langle a_1,a_2,\dots ,a_d \rangle $ $⟨ a_{1}, a_{2}, \dots, a_{d} ⟩$ minimally generated by d coprime positive integers, where $a_1 $ $a_{1}$ denotes the multiplicity of S. In 2015, Moscariello and Sammartano [9] established the validity of Wilf’s inequality for all those numerical semigroups where $a_1$ $a_{1}$ is sufficiently large and all the prime divisors of $a_1$ $a_{1}$ are greater than or equal to $\rho $ $ρ$ , for every fixed value of $\rho $ $ρ$ = $\big \lceil \frac{a_1}{d} \big \rceil .$ $⌈ \frac{a_{1}}{d} ⌉ .$ In this article, we relax the arithmetic constraint on the prime divisors of $a_1$ $a_{1}$ by replacing it with the more natural condition $\gcd (a_1,a_2) = 1,$ $gcd (a_{1}, a_{2}) = 1,$ thereby significantly enlarging the family of numerical semigroups known to satisfy Wilf’s Conjecture. Moreover, under this hypothesis, we sharpen the bound on $a_1$ $a_{1}$ appearing in the work of Moscariello and Sammartano. As a consequence, our results not only extend their theorem to a wider class of numerical semigroups but also demonstrate that Wilf’s inequality holds under strictly improved numerical bounds.