154 problems

Let $S$ be a numerical semigroup. Write $\operatorname{e}(S)$ for its embedding dimension, $\operatorname{n}(S)$ for the number of elements of $S$ below its conductor, and…

Let $\Gamma$ be a symmetric numerical semigroup, let $\Bbbk$ be a field, and let $R=\Bbbk[\Gamma]$ be its semigroup algebra. Let $M$ be a 2-generated ideal of $R$, viewed as an $R$…

Let $n(g)$ denote the number of numerical semigroups of genus $g$. Bras-Amorós' conjecture. For each integer $g\geqslant 1$, … This is a conjecture about the growth of the number o…

Let $n_g$ denote the number of numerical semigroups of genus $g$. The Fibonacci conjecture. The sequence $(n_g)_{g\geq 0}$ is increasing, each term is at least the sum of the two p…

Let $S_0(a,b)$ be the numerical semigroup generated by $a$ and $b$, let $g_{0,a,b}$ be its Frobenius number, and let $\pi_{0,a,b}$ count the primes in $S_0(a,b)$ not exceeding…

Let $S$ be a numerical semigroup, let $I_S$ denote its associated ideal, let $\mu(I_S)$ be the cardinality of a minimal generating set of $I_S$, and let $\operatorname{wd}(S)$ deno…

Let $S$ be the numerical semigroup under consideration, with multiplicity $m$, generators $d_1,\dots,d_m$, gap power sums $G_r(S)$, and invariants $K_p(S)$. Write…

A numerical semigroup $S$ is an additive submonoid of $\mathbb{N}$ such that $\mathbb{N}\setminus S$ is finite; its type $t(S)$ is the number of its pseudo-Frobenius numbers. Suppo…

Type-bound conjecture. The type of $S$ can be at most three.

Let $D$ be a finite subset of the positive integers such that $1\in D$. For a numerical semigroup $S$, let $\Delta_0(S)$ and $\Delta_\infty(S)$ denote its 0-delta set and infinity-…

Positive-exponent bound conjecture. The positive exponent indices satisfy

Ciolan–García-Sánchez–Moree conjecture. A numerical semigroup $S$ is cyclotomic if and only if it is complete intersection.

Let $N_m(g)$ denote the number of numerical semigroups $S$ with multiplicity $\mathsf m(S)=m$ and genus $g(S)=g$. Kaplan's conjecture. For any $m\ge 2$, we have … This refines the…

Fix integers $c,h\geq0$, and for each $n\geq2$ let … Suppose that $\Delta(S_2)=\{1,c_0,c_1,\ldots,c_k\}$. The generalization conjecture. Then … Moreover, for $c\geq4$, $x\geq2$, an…

Arnold's density-profile conjecture. Asymptotically for large ${\bf d}^m$, the density at $s$ is

Arnold's conductor conjecture. The conductor is weakly asymptotic to

The determinantal Betti numbers conjecture. All KW semigroups have determinantal Betti numbers. Consequently, the Betti numbers of KW semigroups depend only on their embedding dime…

Let $a$ be an integer with $a>30$, and let … Let $S(A(a))$ be the numerical semigroup generated by $A(a)$. For $n\in\mathbb{Z}^{+}$, let $\iota(n)$ be the least number of positive…

Inner automorphism conjecture. The automorphisms of $\mathcal P(\mathbb Z)$ are all inner, and the automorphisms of $\mathcal P_0(\mathbb Z)$ are all inner.

Eliahou-number conjecture. If $b'=h$ and $S$ is short, then $E\ge0$.

Let $S$ be a numerical semigroup of multiplicity $m=2k$, where $k\in\mathbb{Z}$. A decomposition of $S$ is an irredundant intersection of irreducible numerical semigroups, and its…

A numerical semigroup is an additive subsemigroup of the nonnegative integers containing $0$ and having finite complement. Every numerical semigroup can be expressed as an irredund…

Fix the multiplicity $e$. Betti-number stability conjecture. For every $m\geq 4$, the Betti numbers of $S^e(m,m+1)$ and $S^e(m+1,m+2)$ agree through the indicated range; equivalent…

Fix the multiplicity $e$ and write $\beta_j(a,b):=\beta_j(S^e(a,b))$. Trailing Betti-number conjecture for S^e(2,n). For $n\geq 6$, … and … Moreover, $S^e(2,e-1)$ is conjectured to…

Fix the multiplicity $e$ and write $\beta_j(m):=\beta_j(S^e(m))$. Trailing Betti-number conjecture. For every $m\geq 2$, the Betti numbers satisfy … and … This proposes that the Be…