arXiv subjects

Caleb McKinley Shor

Publications and source records attributed to Caleb McKinley Shor.

4 recordsLinked to original sources

arXiv2026

Parity distributions among gaps of free numerical semigroups

In this paper, we extend recent results about the distribution of even and odd gaps of a numerical semigroup. We find that, for any numerical semigroup, the distribution can be computed in terms of the numbers of or the sums of odd and even elements in a corresponding Apéry set. With free numerical semigroups specifically, we show that there are always at least as many odd gaps as even gaps, with equality precisely when the generating elements are all odd. We then specialize these results to the cases of numerical semigroups generated by compound and geometric sequences.

math.NT↗

arXiv2022

Equidistribution of numerical semigroup gaps modulo $m$

For a positive integer $m$, a finite set of integers is said to be equidistributed modulo $m$ if the set contains an equal number of elements in each congruence class modulo $m$. In this paper, we consider the problem of determining when the set of gaps of a numerical semigroup $S$ is equidistributed modulo $m$. Of particular interest is the case when the nonzero elements of an Apéry set of $S$ form an arithmetic sequence. We explicitly describe such numerical semigroups $S$ and determine conditions for which the sets of gaps of these numerical semigroups are equidistributed modulo $m$.

math.NT↗

arXiv2018

Characterizations of numerical semigroup complements via Apéry sets

In this paper, we generalize the work of Tuenter to give an identity which completely characterizes the complement of a numerical semigroup in terms of its Apéry sets. Using this result, we compute the $m$th power Sylvester and alternating Sylvester sums for free numerical semigroups. Explicit formulas are given for small $m$.

math.NT↗

arXiv2017

Higher-order Weierstrass weights of branch points on superelliptic curves

In this paper we consider the problem of calculating the higher-order Weierstrass weight of the branch points of a superelliptic curve $C$. For any $q>1$, we give an exact formula for the $q$-weight of an affine branch point. We also find a formula for the $q$-weight of a point at infinity in the case where $n$ and $d$ are relatively prime. With these formulas, for any fixed $n$, we obtain an asymptotic formula for the ratio of the $q$-weight of the branch points, denoted $BW_q$, to the total $q$-weight of points on the curve: \[ \liminf_{d\to\infty}\frac{BW_q}{g(g-1)^2(2q-1)^2}\geq \frac{n+1}{3(n-1)^2(2q-1)^2},\] with equality when the limit is taken such that $\gcd(n,d)=1$.

math.AG↗