arXiv subjects

Bianca Thompson

Publications and source records attributed to Bianca Thompson.

9 recordsLinked to original sources

arXiv2026

Community-driven data science practices

Mathematics researchers are becoming more involved with research questions at the interface of data science and social justice. This type of research needs to be grounded in the needs of the community in order to have significant impact. In this paper, we examine two examples of community-research partnerships in data science for social justice co-authored by both community members and mathematical researchers. The first, VECINA, is a place-based community-research partnership focused on environmental justice. VECINA introduces a framework for developing fruitful local collaborations. The second example, SToPA, originates in citizens' request for an analysis of their town's policing data, but focuses on how to scale this work beyond that place-based setting. SToPA's research helps us imagine how we can continue to actively collaborate with community members even when working to scale projects beyond a single community. In both of these case studies, we examine the harmonies between established principles of power, process, and perspective with our framework for research-community partnerships. We use a duoethnography approach, directly illustrating the experiences of researchers. We also offer a set of reflections on the impact of these research-community partnerships.

math.HO↗

arXiv2024

Dynamical Irreducibility of Certain Families of Polynomials over Finite Fields

We determine necessary and sufficient conditions for unicritical polynomials to be dynamically irreducible over finite fields. This result extends the results of Boston-Jones and Hamblen-Jones-Madhu regarding the dynamical irreducibility of particular families of unicritical polynomials. We also investigate dynamical irreducibility conditions for cubic and shifted linearized polynomials.

math.NT↗

arXiv2022

Minimum Distance and Parameter Ranges of Locally Recoverable Codes with Availability from Fiber Products of Curves

We construct families of locally recoverable codes with availability $t\geq 2$ using fiber products of curves, determine the exact minimum distance of many families, and prove a general theorem for minimum distance of such codes. The paper concludes with an exploration of parameters of codes from these families and the fiber product construction more generally. We show that fiber product codes can achieve arbitrarily large rate and arbitrarily small relative defect, and compare to known bounds and important constructions from the literature.

cs.IT↗

arXiv2021

Solving Quadratic and Cubic Diophantine Equations using 2-adic Valuation Trees

For fixed integers $D \geq 0$ and $c \geq 3$, we demonstrate how to use $2$-adic valuation trees of sequences to analyze Diophantine equations of the form $x^2+D=2^cy$ and $x^3+D=2^cy$, for $y$ odd. Further, we show for what values $D \in \mathbb{Z}^+$, the numbers $x^3+D$ will generate infinite valuation trees, which lead to infinite solutions to the above Diophantine equations.

math.NT↗

arXiv2021

Density of Periodic Points for Latt\`es maps over Finite Fields

Let $L_d$ be the Latt\`es map associated to the multiplication-by-$d$ endomorphism of an elliptic curve $E$ defined over a finite field $\mathbb{F}_q$. We determine the density $\delta(L_d,q)$ of periodic points for $L_d$ in $\mathbb{P}^1(\mathbb{F}_q)$. We show that the periodic point densities $\delta(L_d,q^n)$ converge as $n \rightarrow \infty$ along certain arithmetic progressions, and compute simple explicit formulas for $\delta(L_\ell,q)$ when $\ell$ is a prime and $E$ belongs to a special family of supersingular elliptic curves.

math.NT↗

arXiv2018

Arboreal representations for rational maps with few critical points

Jones conjectures the arboreal representation of a degree two rational map will have finite index in the full automorphism group of a binary rooted tree except under certain conditions. We prove a version of Jones' Conjecture for quadratic and cubic polynomials assuming the $abc$-Conjecture and Vojta's Conjecture. We also exhibit a family of degree $2$ rational maps and give examples of degree $3$ polynomial maps whose arboreal representations have finite index in the appropriate group of tree automorphisms.

math.NT↗

arXiv2016

On the field of definition of a cubic rational function and its critical points

Using essentially only algebra, we give a proof that a cubic rational function over $\mathbb{C}$ with real critical points is equivalent to a real rational function. We also show that the natural generalization to $\mathbb{Q}_p$ fails for all $p$.

math.NT↗

arXiv2013

Periodic points in towers of finite fields for polynomials associated to algebraic groups

We find the limiting proportion of periodic points in towers of finite fields for polynomial maps associated to algebraic groups, namely pure power maps z^d and Chebyshev polynomials.

math.NT↗

arXiv2012

Uniform bounds for pre-periodic points in families of twists

Let phi be a morphism of projective N-space defined over a number field K. We prove that there is a bound B depending only on phi such that every twist of phi has no more than B K-rational preperiodic points. (This result is analagous to a result of Silverman for abelian varieties.) For two specific families of quadratic rational maps over Q, we find the bound B explicitly.

math.NT↗