arXiv subjects

Robin Visser

Publications and source records attributed to Robin Visser.

7 recordsLinked to original sources

arXiv2026

On integers that are representable as the sum of two units

Let $K$ be a number field of degree $D$ with maximal order $\mathcal{O}_K$. We show that under certain conditions on $K$, which in particular are always satisfied if $D$ is odd or if $D \geq 3$ and $K$ is primitive, the set of positive integers $N_K$ that can be expressed as a sum of two units in $\mathcal{O}_K^*$ is a finite effectively computable set. This result partially resolves an open problem posed by Tinkov\'{a}, Yatsyna, and the first author. We illustrate our method by explicitly computing $N_K$ for the smallest totally real quintic field $K$ with Galois group $S_5$.

math.NT↗

arXiv2026

Repellent properties of perfect powers on partition functions: a heuristic approach

In 2013, Sun conjectured that the partition function $p(n)$ is never a perfect power for $n \geq 2$. Building on this, Merca, Ono, and Tsai recently observed that for any fixed integers $d \geq 0$ and $k \geq 2$, there appear to be only finitely many integers $n$ such that $p(n)$ differs from a perfect $k$th power by at most $d$. Denoting by $M_k(d)$ the largest such $n$, they conjectured that $M_k(d) = o(d^\epsilon)$ for every $\epsilon > 0$. In this paper, we investigate the asymptotic growth of analogs of $M_k(d)$ for a wide class of partition functions. We establish sharp lower bounds and provide heuristics which suggest that $M_k(d)$ in fact grows polylogarithmically in $d$, i.e. of order $\log^2(d)$. More generally, we prove that if $f(n)$ is a suitably random chosen function with asymptotic growth rate similar to that of $p(n)$, then the set of integers $n$ for which $f(n)$ is a perfect power is finite with probability 1.

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arXiv2025

Asymptotics of $n$-universal lattices over number fields

We prove an explicit asymptotic formula for the logarithm of the minimal ranks of $n$-universal lattices over the ring of integers of totally real number fields. We also show that, for any constant $C > 0$ and $n \geq 3$, there are only finitely many totally real fields with an $n$-universal lattice of rank at most $C$, with all such fields being effectively computable. Similarly, for any $n \geq 3$, we show that there are only finitely many totally real fields admitting an $n$-universal criterion set of size at most $C$, with all such fields likewise being effectively computable.

math.NT↗

arXiv2025

Rectangle partitions generalizing integer partitions

In this paper, we introduce a natural geometric extension of the partition function. More precisely, we investigate the problem of counting partitions of a rectangle into rectangular blocks with integer sides. Here, two partitions of a rectangle are indistinguishable if they consist of the same multiset of blocks, their geometric arrangement does not matter.

math.CO↗

arXiv2025

Sums of two units in number fields

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathcal{N}_K$ be the set of positive integers $n$ such that there exist units $\varepsilon, \delta \in \mathcal{O}_K^\times$ satisfying $\varepsilon + \delta = n$. We show that $\mathcal{N}_K$ is a finite set if $K$ does not contain any real quadratic subfield. In the case where $K$ is a cubic field, we also explicitly classify all solutions to the unit equation $\varepsilon + \delta = n$ when $K$ is either cyclic or has negative discriminant.

math.NT↗

arXiv2023

Curves with few bad primes over cyclotomic $\mathbb{Z}_\ell$-extensions

Let $K$ be a number field, and $S$ a finite set of non-archimedean places of $K$, and write $\mathcal{O}_S^\times$ for the group of $S$-units of $K$. A famous theorem of Siegel asserts that the $S$-unit equation $\varepsilon+\delta=1$, with $\varepsilon$, $\delta \in \mathcal{O}_S^\times$, has only finitely many solutions. A famous theorem of Shafarevich asserts that there are only finitely many isomorphism classes of elliptic curves over $K$ with good reduction outside $S$. Now instead of a number field, let $K=\mathbb{Q}_{\infty,\ell}$ which denotes the $\mathbb{Z}_\ell$-cyclotomic extension of $\mathbb{Q}$. We show that the $S$-unit equation $\varepsilon+\delta=1$, with $\varepsilon$, $\delta \in \mathcal{O}_S^\times$, has infinitely many solutions for $\ell \in \{2,3,5,7\}$, where $S$ consists only of the totally ramified prime above $\ell$. Moreover, for every prime $\ell$, we construct infinitely many elliptic or hyperelliptic curves defined over $K$ with good reduction away from $2$ and $\ell$. For certain primes $\ell$ we show that the Jacobians of these curves in fact belong to infinitely many distinct isogeny classes.

math.NT↗

arXiv2022

Potential good reduction of hyperelliptic curves

Let $K$ be a number field, and $g \geq 2$ a positive integer. We define $c_K(g)$ as the smallest integer $n$ such that there exist infinitely many $\overline{K}$-isomorphism classes of genus $g$ hyperelliptic curves $C/K$ with all Weierstrass points in $K$ having potentially good reduction outside $n$ primes in $K$. We show that $c_K(g) > \pi_{K, \textrm{odd}}(2g) + 1$, where $\pi_{K, \textrm{odd}}(n)$ denotes the number of odd primes in $K$ with norm no greater than $n$, as well as present a summary of various conditional and unconditional results on upper bounds for $c_K(g)$.

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