Given coprime positive integers g(1) &lt; ... &lt; g(e), the Frobenius number F = F(g(1),..., g(e)) is the largest integer not representable as a linear combination of g(1),..., g(e) with non-negative integer coefficients. Let n denote the number of all representable non-negative integers less than F; Wilf conjectured that F + 1 &lt;= en. We provide bounds for g1 and for the type of the numerical semigroup S = &lt; g(1),..., g(e)&gt; in function of e and n, and use these bounds to prove that F + 1 &lt;= qen, where q = [F+1/g1], and F + 1 &lt;= en(2). Finally, we give an alternative, simpler proof for theWilf conjecture if the numerical semigroup S = &lt; g(1),..., g(e)&gt; is almost-symmetric.

### Bounds for invariants of numerical semigroups and Wilf's conjecture

#### Abstract

Given coprime positive integers g(1) < ... < g(e), the Frobenius number F = F(g(1),..., g(e)) is the largest integer not representable as a linear combination of g(1),..., g(e) with non-negative integer coefficients. Let n denote the number of all representable non-negative integers less than F; Wilf conjectured that F + 1 <= en. We provide bounds for g1 and for the type of the numerical semigroup S = < g(1),..., g(e)> in function of e and n, and use these bounds to prove that F + 1 <= qen, where q = [F+1/g1], and F + 1 <= en(2). Finally, we give an alternative, simpler proof for theWilf conjecture if the numerical semigroup S = < g(1),..., g(e)> is almost-symmetric.
##### Scheda breve Scheda completa Scheda completa (DC)
2023
Wilf conjecture
Numerical semigroups
Multiplicity
Embedding dimension
Type
Almost symmetric numerical semigroup
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/20.500.11769/577089`