arXiv subjects

Elisa Bellah

Publications and source records attributed to Elisa Bellah.

5 recordsLinked to original sources

arXiv2025

Connectedness of special points in the Markoff mod $p$ graphs

It is conjectured that the Markoff equation $X^2+Y^2+Z^2=3XYZ$ satisfies the special Diophantine property that every mod $p$ solution lifts to an integer solution. Progress toward this conjecture has been made by studying the connectedness of the graphs obtained from the action of the Vieta group on the nonzero mod $p$ solutions to the Markoff equation. In this paper, we use results on Pisano periods of the Fibonacci sequence to obtain explicit results on the connectedness of special points in this graph for primes $p$ where $p+1$ has large $2$-adic valuation. In particular, for Mersenne primes $p \equiv \pm 2 \,(\text{mod}\, 5)$, we show that the special point $(1, 1, 1)$ which is fixed under reduction modulo $p$ lies in a component of this graph which is known to be connected.

math.NT↗

arXiv2023

Bounding Lifts of Markoff Triples mod $p$

In 2016, Bourgain, Gamburd, and Sarnak proved that Strong Approximation holds for the Markoff surface in most cases. That is, the modulo $p$ solutions to the equation $X_1^2+X_2^2+X_3^2=3X_1X_2X_3$ are covered by the integer solutions for most primes $p$. In this paper, we provide upper bounds on lifts of mod $p$ points of the Markoff surface by analyzing the growth along paths in the Markoff mod $p$ graphs. Our first upper bound follows the algorithm given in the paper of Bourgain, Gamburd, and Sarnak, which constructs a path of possibly long length but where points grow relatively slowly. Our second bound considers paths in these graphs of short length but possibly large growth. We then provide numerical evidence and heuristic arguments for how these bounds might be improved.

math.NT↗

arXiv2023

Arbitrary finite intersections of doubling measures and applications

Using a wide array of machinery from diverse fields across mathematics, we provide a construction of a measure on the real line which is doubling on all $n$-adic intervals for any finite list of $n\in\mathbb{N}$, yet not doubling overall. In particular, we extend previous results in the area, where only two coprime numbers $n$ were allowed, by using substantially new ideas. In addition, we provide several nontrivial applications to reverse Hölder weights, $A_p$ weights, Hardy spaces, BMO and VMO function classes, and connect our results with key principles and conjectures across number theory.

math.NT↗

arXiv2021

Norm Form Equations and Linear Divisibility Sequences

Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It is known that these solutions can be written as tuples of linear recurrence sequences. We show that for certain families of norm forms defined over quartic fields, there exist integrally equivalent forms making any one fixed coordinate sequence a linear divisibility sequence.

math.NT↗

arXiv2016

A probabilistic heuristic for counting components of functional graphs of polynomials over finite fields

In 2014, Flynn and the second author bounded the average number of components of the functional graphs of polynomials of fixed degree over a finite field. When the fixed degree was large (relative to the size of the finite field), their lower bound matched Kruskal's asymptotic for random functional graphs. However, when the fixed degree was small, they were unable to match Krusal's bound, since they could not (Lagrange) interpolate cycles in functional graphs of length greater than the fixed degree. In our work, we introduce a heuristic for approximating the average number of such cycles of any length. This heuristic is, roughly, that for sets of edges in a functional graph, the quality of being a cycle and the quality of being interpolable are "uncorrelated enough". We prove that this heuristic implies that the average number of components of the functional graphs of polynomials of fixed degree over a finite field is within a bounded constant of Kruskal's bound. We also analyze some numerical data comparing implications of this heuristic to some component counts of functional graphs of polynomials over finite fields.

math.DS↗