92 problems

For every numerical semigroup $H=\langle a_1,\ldots,a_n\rangle$ minimally generated by $n$ elements, let $I_H=\ker\bigl(k[x_1,\ldots,x_n]\to k[t]\bigr)$, where $x_i\mapsto t^{a_i}$…

Let $H=\langle a_1,\ldots,a_n\rangle$ be a numerical semigroup minimally generated by $a_1,\ldots,a_n$, let $S=k[x_1,\ldots,x_n]$, and let…

Does there exist a function $f:\mathbb{N}\to\mathbb{N}$ such that, for every numerical semigroup $S$ with embedding dimension $4$, the type satisfies $t(S)\le f\bigl(\eta(S)\bigr)$…

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…

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 $n_{g,r}$ denote the number of numerical semigroups of genus $g$ with ordinarization number $r$, where the ordinarization number is the length of the path from the semigroup to…

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…

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

Let $S\subseteq\mathbb{N}^{d}$ be a Frobenius generalized numerical semigroup with Frobenius element $\mathbf{f}=(f^{(1)},\ldots,f^{(d)})$. With $e(S)$ the number of minimal genera…

Convergence conjecture. The sum in the displayed series converges to a finite value.

Let $a\geqslant 3$ be an odd integer, and let $\langle a,a+2\rangle$ denote the numerical semigroup generated by $a$ and $a+2$. Let $(u_n)_{n\geqslant 1}$ be the recursive sequence…

Let $m\geq 3$ and consider the regions arising in Algorithm for testing numerical semigroups of multiplicity $m$. Empty-region conjecture. For each $m\geq 3$, every region consider…

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

Arnold's average-density conjecture. For large ${\bf d}^m$, with overwhelming probability, this fraction is asymptotically $1/m$; equivalently,

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.

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…

Tringali–Yan conjecture. If $S$ is a numerical monoid properly contained in $\mathbb{N}$, then the automorphism group of $\mathcal{P}_{0}(S)$ is the trivial group:

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…

Let $G_0(a,b)$ be the set of gaps of the numerical semigroup generated by $a$ and $b$, let $g_{0,a,b}$ be its Frobenius number, and let $\pi^*_{0,a,b}$ be the number of primes in…

Let $n(g,t)$ denote the number of numerical semigroups of genus $g$ and type $t$. Type-monotonicity conjecture. For each pair of integers satisfying $2\leqslant t\leqslant g$, … Th…

Let $S$ be a numerical semigroup, and call a nonzero small element of $S$ lonely if it has the property specified in the source. Lonely element conjecture. If every nonzero small e…

Let $a$ and $b$ be integers with $b>a\ge 1$ and $\gcd(a,b)=1$. Set $S(a,b)=ab-a-b$, and let $\pi^*(a,b)$ be the number of primes not belonging to the numerical semigroup…