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Wilf's question in numerical semigroups $S_3$ revisited and inequalities for rescaled genera

Leonid G. Fel

TL;DR

The paper investigates Wilf's conjecture for numerical semigroups generated by three elements $S_3$, employing syzygy-degree identities to derive sharp bounds on key invariants. By translating polynomial identities for syzygy degrees into power-sum and symmetric-polynomial inequalities, the authors obtain a lower bound for the Frobenius number via $g_3$ and explicit bounds for the rescaled genera $h_r=K_r/g_3^{r+1}$, including concrete values for $h_1,h_2,h_3$ and a general algebraic relation for higher $h_r$. They prove an affirmative answer to Wilf's question for all non-symmetric $S_3$ and provide a finite-band Diophantine check that reinforces this result; they also analyze symmetric (complete intersection) cases, giving explicit $h_r$ in terms of a parameter $v$ and showing these bounds lie within the non-symmetric ranges. The work highlights a method that could extend to $S_m$ with $m>3$ and offers a framework for bounding rescaled genera across families of semigroups, with implications for the structure of gaps, conductor, and genus. Overall, the paper advances Wilf-type bounds in the $3$-generator case and deepens understanding of how syzygies constrain semigroup invariants.

Abstract

We consider numerical semigroups $S_3 = \langle d_1,d_2,d_3\rangle$, minimally generated by three positive integers. We revisit the Wilf question in $S_3$ and, making use of identities for degrees of syzygies of such semigroups, give a short proof of existence of an affirmative answer. We find also the lower bound for Frobenius numbers of $S_3$ and upper and lower bounds for rescaled genera.

Wilf's question in numerical semigroups $S_3$ revisited and inequalities for rescaled genera

TL;DR

The paper investigates Wilf's conjecture for numerical semigroups generated by three elements , employing syzygy-degree identities to derive sharp bounds on key invariants. By translating polynomial identities for syzygy degrees into power-sum and symmetric-polynomial inequalities, the authors obtain a lower bound for the Frobenius number via and explicit bounds for the rescaled genera , including concrete values for and a general algebraic relation for higher . They prove an affirmative answer to Wilf's question for all non-symmetric and provide a finite-band Diophantine check that reinforces this result; they also analyze symmetric (complete intersection) cases, giving explicit in terms of a parameter and showing these bounds lie within the non-symmetric ranges. The work highlights a method that could extend to with and offers a framework for bounding rescaled genera across families of semigroups, with implications for the structure of gaps, conductor, and genus. Overall, the paper advances Wilf-type bounds in the -generator case and deepens understanding of how syzygies constrain semigroup invariants.

Abstract

We consider numerical semigroups , minimally generated by three positive integers. We revisit the Wilf question in and, making use of identities for degrees of syzygies of such semigroups, give a short proof of existence of an affirmative answer. We find also the lower bound for Frobenius numbers of and upper and lower bounds for rescaled genera.
Paper Structure (7 sections, 1 theorem, 46 equations, 1 figure)

This paper contains 7 sections, 1 theorem, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Syzygy identities for numerical semigroups $S_3$
  3. Lower bound of $g_3$
  4. Lower and upper bounds of $h_0=K_0/g_3$
  5. Wilf's question in $S_3$.
  6. Lower and upper bounds of $h_r=K_r/g_3^{r+1}$, $r\ge 1$.
  7. Rescaled genera $h_r$ in symmetric semigroups $S_3$

Key Result

Theorem 1

Let a non-symmetric numerical semigroup $\langle d_1, d_2,d_3\rangle$, $d_1\ge 4$, be given and its generators satisfy (a1). The WQ has an affirmative answer for all numerical semigroups $S_3$

Figures (1)

  • Figure 1: Plot of the function $\Phi(u,v)\!\!=\!\!\frac{1}{2}\left(u\!-\sqrt{Q(u,v)}\right)$ and the points $(u,v,h_0)$ for semigroups $\langle 3,4,5\rangle$ ( yellow), $\langle 4,5,6\rangle$ ( purple), $\langle 5,6,7\rangle$ ( blue), $\langle 11,17,29\rangle$ ( red), $\langle 25,31,43\rangle$ ( magenta), $\langle 23,29,44\rangle$ ( cyan), $\langle 43,47,113\rangle$ ( green), $\langle 501,503,603\rangle$ ( black), $\langle 901,903,1003\rangle$ ( gray) and $\langle 1201,1203,1303\rangle$ ( orange).

Theorems & Definitions (1)

  • Theorem 1