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Ariane M. Masuda

Publications and source records attributed to Ariane M. Masuda.

17 recordsLinked to original sources
arXiv2026

QEDBENCH: Quantifying the Alignment Gap in Automated Evaluation of University-Level Mathematical Proofs

As Large Language Models (LLMs) saturate elementary benchmarks, the research frontier has shifted from generation to the reliability of automated evaluation. We demonstrate that standard "LLM-as-a-Judge" protocols suffer from a systematic Alignment Gap when applied to upper-undergraduate to early graduate level mathematics. To quantify this, we introduce QEDBench, the first large-scale dual-rubric alignment benchmark to systematically measure alignment with human experts on university-level math proofs by contrasting course-specific rubrics against expert common knowledge criteria. By deploying a dual-evaluation matrix (7 judges x 5 solvers) against 1,000+ hours of human evaluation, we reveal that certain frontier evaluators like Claude Opus 4.5, DeepSeek-V3, Qwen 2.5 Max, and Llama 4 Maverick exhibit significant positive bias (up to +0.18, +0.20, +0.30, +0.36 mean score inflation, respectively). Furthermore, we uncover a critical reasoning gap in the discrete domain: while Gemini 3.0 Pro achieves state-of-the-art performance (0.91 average human evaluation score), other reasoning models like GPT-5 Pro and Claude Sonnet 4.5 see their performance significantly degrade in discrete domains. Specifically, their average human evaluation scores drop to 0.72 and 0.63 in Discrete Math, and to 0.74 and 0.50 in Graph Theory. In addition to these research results, we also release QEDBench as a public benchmark for evaluating and improving AI judges. Our benchmark is publicly published at https://github.com/qqliu/Yale-QEDBench.

cs.LG↗
arXiv2023

Solving the membership problem for certain subgroups of $SL_2(\mathbb{Z})$

For positive integers $u$ and $v$, let $L_u=\begin{bmatrix}1 & 0 \\u&1\end{bmatrix}$ and $R_v=\begin{bmatrix}1 & v \\ 0 & 1\end{bmatrix}$. Let $G_{u,v}$ be the group generated by $L_u$ and $R_v$. In a previous paper, the authors determined a characterization of matrices $M=\begin{bmatrix}a & c \\b&d\end{bmatrix}$ in $G_{u,v}$ when $u,v\geq 3$ in terms of the short continued fraction representation of $b/d$. We extend this result to the case where $u+v> 4$. Additionally, we compute $[\mathscr{G}_{u,v}\colon G_{u,v}]$ for $u,v\geq 1$, extending a result of Chorna, Geller, and Shpilrain.

math.GR↗
arXiv2022

Rédei permutations with the same cycle structure

Let $\mathbb{F}_q$ be the finite field of order $q$, and $\mathbb P^1(\mathbb{F}_q) = \mathbb F_q\cup \{\infty\}$. Write $(x+\sqrt y)^m$ as $N(x,y)+D(x,y)\sqrt{y}$. For $m\in\mathbb N$ and $a \in \mathbb{F}_q$, the Rédei function $R_{m,a}\colon \mathbb P^1(\mathbb F_q) \to \mathbb P^1(\mathbb F_q)$ is defined by $N(x,a)/D(x,a)$ if $D(x,a)\neq 0$ and $x\neq\infty$, and $\infty$, otherwise. In this paper we give a complete characterization of all pairs $(m,n)\in\mathbb N^2$ such that the Rédei permutations $R_{m,a}$ and $R_{n,b}$ have the same cycle structure when $a$ and $b$ have the same quadratic character and $q$ is odd. We explore some relationships between such pairs $(m,n)$, and provide explicit families of Rédei permutations with the same cycle structure. When a Rédei permutation has a unique cycle structure that is not shared by any other Rédei permutation, we call it isolated. We show that the only isolated Rédei permutations are the isolated Rédei involutions. Moreover, all our results can be transferred to bijections of the form $mx$ and $x^m$ on certain domains.

math.NT↗
arXiv2022

Permutation binomials of the form $x^r(x^{q-1}+a)$ over $\mathbb F_{q^e}$

We present several existence and nonexistence results for permutation binomials of the form $x^r(x^{q-1}+a)$, where $e\geq 2$ and $a\in \mathbb{F}_{q^e}^*$. As a consequence, we obtain a complete characterization of such permutation binomials over $\mathbb{F}_{q^2}$, $\mathbb{F}_{q^3}$, $\mathbb{F}_{q^4}$, $\mathbb{F}_{p^5}$, and $\mathbb{F}_{p^6}$, where $p$ is an odd prime.

math.NT↗
arXiv2020

Rédei permutations with cycles of the same length

Let $\mathbb{F}_q$ be a finite field of odd characteristic. We study Rédei functions that induce permutations over $\mathbb{P}^1(\mathbb{F}_q)$ whose cycle decomposition contains only cycles of length $1$ and $j$, for an integer $j\geq 2$. When $j$ is $4$ or a prime number, we give necessary and sufficient conditions for a Rédei permutation of this type to exist over $\mathbb{P}^1(\mathbb{F}_q)$, characterize Rédei permutations consisting of $1$- and $j$-cycles, and determine their total number. We also present explicit formulas for Rédei involutions based on the number of fixed points, and procedures to construct Rédei permutations with a prescribed number of fixed points and $j$-cycles for $j \in \{3,4,5\}$.

math.NT↗
arXiv2020

Maximal entries of elements in certain matrix monoids

Let $L_u=\begin{bmatrix}1 & 0\\u & 1\end{bmatrix}$ and $R_v=\begin{bmatrix}1 & v\\0 & 1\end{bmatrix}$ be matrices in $SL_2(\mathbb Z)$ with $u, v\geq 1$. Since the monoid generated by $L_u$ and $R_v$ is free, we can associate a depth to each element based on its product representation. In the cases where $u=v=2$ and $u=v=3$, Bromberg, Shpilrain, and Vdovina determined the depth $n$ matrices containing the maximal entry for each $n\geq 1$. By using ideas from our previous work on $(u,v)$-Calkin-Wilf trees, we extend their results for any $u, v\geq 1$ and in the process we recover the Fibonacci and some Lucas sequences. As a consequence we obtain bounds which guarantee collision resistance on a family of hashing functions based on $L_u$ and $R_v$.

math.NT↗
arXiv2020

Subgroups of $SL_2(\mathbb{Z})$ characterized by certain continued fraction representations

For positive integers $u$ and $v$, let $L_u=\begin{bmatrix} 1 & 0 \\ u & 1 \end{bmatrix}$ and $R_v=\begin{bmatrix} 1 & v \\ 0 & 1 \end{bmatrix}$. Let $S_{u,v}$ be the monoid generated by $L_u$ and $R_v$, and $G_{u,v}$ be the group generated by $L_u$ and $R_v$. In this paper we expand on a characterization of matrices $M=\begin{bmatrix}a & b \\c & d\end{bmatrix}$ in $S_{k,k}$ and $G_{k,k}$ when $k\geq 2$ given by Esbelin and Gutan to $S_{u,v}$ when $u,v\geq 2$ and $G_{u,v}$ when $u,v\geq 3$. We give a simple algorithmic way of determining if $M$ is in $G_{u,v}$ using a recursive function and the short continued fraction representation of $b/d$.

math.GR↗
arXiv2018

Mean Row Values in $(u,v)$-Calkin-Wilf Trees

We fix integers $u,v \geq 1$, and consider an infinite binary tree $\mathcal{T}^{(u,v)}(z)$ with a root node whose value is a positive rational number $z$. For every vertex $a/b$, we label the left child as $a/(ua+b)$ and right child as $(a+vb)/b$. The resulting tree is known as the $(u,v)$-Calkin-Wilf tree. As $z$ runs over $[1/u,v]\cap \mathbb{Q}$, the vertex sets of $\mathcal{T}^{(u,v)}(z)$ form a partition of $\mathbb{Q}^+$. When $u=v=1$, the mean row value converges to $3/2$ as the row depth increases. Our goal is to extend this result for any $u,v\geq 1$. We show that, when $z\in [1/u,v]\cap \mathbb{Q}$, the mean row value in $\mathcal{T}^{(u,v)}(z)$ converges to a value close to $v+\log 2/u$ uniformly on $z$.

math.NT↗
arXiv2018

Pure gaps on curves with many rational places

We consider the algebraic curve defined by $y^m = f(x)$ where $m \geq 2$ and $f(x)$ is a rational function over $\mathbb{F}_q$. We extend the concept of pure gap to {\bf c}-gap and obtain a criterion to decide when an $s$-tuple is a {\bf c}-gap at $s$ rational places on the curve. As an application, we obtain many families of pure gaps at two rational places on curves with many rational places.

math.CO↗
arXiv2018

Permutation polynomials over $\mathbb{F}_{q^2}$ from rational functions

Let $μ_{q+1}$ denote the set of $(q+1)$-th roots of unity in $\mathbb{F}_{q^2 }$. We construct permutation polynomials over $\mathbb{F}_{q^2}$ by using rational functions of any degree that induce bijections either on $μ_{q+1}$ or between $μ_{q+1}$ and $\mathbb{F}_q \cup \{\infty\}$. In particular, we generalize results from Zieve.

math.CO↗
arXiv2016

One- and Two-Point Codes over Kummer Extensions

We compute the Weierstrass semigroup at one totally ramified place for Kummer extensions defined by $y^m=f(x)^λ$ where $f(x)$ is a separable polynomial over $\mathbb{F}_q$. In addition, we compute the Weierstrass semigroup at two certain totally ramified places. We then apply our results to construct one- and two-point algebraic geometric codes with good parameters.

math.AG↗
arXiv2015

The (u,v)-Calkin-Wilf Forest

In this paper we consider a refinement, due to Nathanson, of the Calkin-Wilf tree. In particular, we study the properties of such trees associated with the matrices $L_u=\begin{bmatrix} 1 & 0 \\ u & 1\end{bmatrix}$ and $R_v=\begin{bmatrix} 1 & v \\ 0& 1\end{bmatrix}$, where $u$ and $v$ are nonnegative integers. We extend several known results of the original Calkin-Wilf tree, including the symmetry, numerator-denominator, and successor formulas, to this new setting. Additionally, we study the ancestry of a rational number appearing in a generalized Calkin-Wilf tree.

math.NT↗
arXiv2014

Character Theory of Monoids over an Arbitrary Field

The basic character theory of finite monoids over the complex numbers was developed in the sixties and seventies based on work of Munn, Ponizovsky, McAlister, Rhodes and Zalcstein. In particular, McAlister determined the space of functions spanned by the irreducible characters of a finite monoid over $\mathbb C$ and the ring of virtual characters. In this paper, we present the corresponding results over an arbitrary field. As a consequence, we obtain a quick proof of the theorem of Berstel and Reutenauer that the characteristic function of a regular cyclic language is a virtual character of the free monoid. This is a crucial ingredient in their proof of the rationality of the zeta function of a sofic shift in symbolic dynamics.

math.RT↗
arXiv2008

Permutation binomials over finite fields

We prove that if x^m + c*x^n permutes the prime field GF(p), where m>n>0 and c is in GF(p)^*, then gcd(m-n,p-1) > sqrt{p} - 1. Conversely, we prove that if q>=4 and m>n>0 are fixed and satisfy gcd(m-n,q-1) > 2q*(log log q)/(log q), then there exist permutation binomials over GF(q) of the form x^m + c*x^n if and only if gcd(m,n,q-1) = 1.

math.NT↗
arXiv2007

Nonexistence of permutation binomials of certain shapes

Suppose x^m + c*x^n is a permutation polynomial over GF(p), where p>5 is prime, m>n>0, and c is in GF(p)^*. We prove that gcd(m-n,p-1) is not 2 or 4. In the special case that either (p-1)/2 or (p-1)/4 is prime, this was conjectured in a recent paper by Masuda, Panario and Wang.

math.NT↗
arXiv2007

Rational functions with linear relations

We find all polynomials f,g,h over a field K such that g and h are linear and f(g(x))=h(f(x)). We also solve the same problem for rational functions f,g,h, in case the field K is algebraically closed.

math.NT↗