arXiv subjects

Sophie Zhu

Publications and source records attributed to Sophie Zhu.

3 recordsLinked to original sources

arXiv2026

Connected components of the ranges of twisted divisor functions on number fields

Let $r\in\mathbb{C}$, let $K$ be a finite extension of $\mathbb{Q}$, let $I_K$ be the monoid of integral ideals in the ring of integers $\mathcal{O}_K$ of $K$, and let $χ$ be a Dirichlet character. Then define the twisted ideal divisor function $σ_{r, K, χ} : I_K \rightarrow \mathbb{C}$ by $$σ_{r,K,χ}(I) = \sum_{J \mid I} N(J)^{-r}χ(N(J)),$$ where $N$ denotes the ideal norm. For real $r>1,$ we study the number of connected components $C_{r, K, χ}$ of the closure $\overline{σ_{r,K,χ}(I_K)}$, writing $C_{r,K}$ when $χ$ is the principal character modulo 1. We prove that $C_{r,K,χ}$ is finite when $χ$ is real-valued. When $K = \mathbb{Q}$, we show that for fixed $r > 1,$ every sufficiently large positive integer is realized as $C_{r,\mathbb{Q},χ},$ and if $r$ is sufficiently large, then every positive integer is realized as $χ$ varies. For finite Galois extensions $K$ over $\mathbb{Q}$, we exhibit new exponential lower bounds for $C_{r,K},$ and we prove that for every fixed integer $s \geq 2$, the values $C_{r,K}$ are unbounded as $K$ ranges over degree-$s$ extensions of $\mathbb{Q}$.

math.NT↗

arXiv2022

On the primality and elasticity of algebraic valuations of cyclic free semirings

A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. Under certain mild conditions on a positive algebraic number $α$, the additive monoid $M_α$ of the evaluation semiring $\mathbb{N}_0[α]$ is atomic. The atomic structure of both the additive and the multiplicative monoids of $\mathbb{N}_0[α]$ has been the subject of several recent papers. Here we focus on the monoids $M_α$, and we study its omega-primality and elasticity, aiming to better understand some fundamental questions about their atomic decompositions. We prove that when $α$ is less than 1, the atoms of $M_α$ are as far from being prime as they can possibly be. Then we establish some results about the elasticity of $M_α$, including that when $α$ is rational, the elasticity of $M_α$ is full (this was previously conjectured by S. T. Chapman, F. Gotti, and M. Gotti).

math.AC↗

arXiv2021

Factorizations in evaluation monoids of Laurent semirings

For a positive real number $α$, let $\mathbb{N}_0[α,α^{-1}]$ be the semiring of all real numbers $f(α)$ for $f(x)$ lying in $\mathbb{N}_0[x,x^{-1}]$, which is the semiring of all Laurent polynomials over the set of nonnegative integers $\mathbb{N}_0$. In this paper, we study various factorization properties of the additive structure of $\mathbb{N}_0[α, α^{-1}]$. We characterize when $\mathbb{N}_0[α, α^{-1}]$ is atomic. Then we characterize when $\mathbb{N}_0[α, α^{-1}]$ satisfies the ascending chain condition on principal ideals in terms of certain well-studied factorization properties. Finally, we characterize when $\mathbb{N}_0[α, α^{-1}]$ satisfies the unique factorization property and show that, when this is not the case, $\mathbb{N}_0[α, α^{-1}]$ has infinite elasticity.

math.AC↗