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Sina Saleh

Publications and source records attributed to Sina Saleh.

7 recordsLinked to original sources
arXiv2026

Degree Growth of Iterates of Curves and Likely Intersections

We study the growth of the bidegree of an ample irreducible curve in $\mathbb{P}^1 \times \mathbb{P}^1$ under a product polynomial endomorphism $φ=(f,g)$, where at least one of $f$ and $g$ is non-exceptional. We prove that, if the curve $C$ is not preperiodic under $(f^a,g^b)$ for any $a,b\geq 1$, then the bidegree of $φ^n(C)$ is asymptotic to $(°(g)^n,°(f)^n)$. As an application of this exponential growth, we prove a geometric analogue of Silverman's theorem on the finiteness of $S$-integral points in orbits. Namely if $C$ is not $(f^a,g^b)$-preperiodic and $C'$ is not totally invariant for $φ$, then for any infinite sequence ${n_i}$ of positive integers, the union of the intersections $$ \bigcup_{i \geq 1} \left( φ^{n_i}(C)\cap C' \right) $$ is Zariski dense in $C'$.

math.DS↗
arXiv2026

Bialgebraic geometry of Böttcher coordinates

Becker and Bergweiler showed that if $f$ is a non-exceptional polynomial, then the Böttcher coordinate $Ψ_f \colon \mathbb D_R \to B_\infty(f)$ associated to $f$ is a transcendental function. In this paper, we study $f$-bialgebraic sets: algebraic subsets of $\mathbb D_R^n$ whose image under the coordinate-wise action of $Ψ_f$ is contained in an algebraic set of the same dimension. We give a complete dynamical classification of bialgebraic sets under the additional assumption that the Julia set of $f$ is either disconnected, or connected and admits a nondegenerate locally connected model. Inspired by the Ax--Lindemann--Weierstrass theorem and the Ax--Schanuel conjecture, we formulate analogs with $Ψ_f$ in place of the exponential function and prove them in the case where the Julia set $J_f$ is disconnected.

math.DS↗
arXiv2025

Rational self-maps of projective surfaces with a regular iterate

We show that if $Φ: X \dashrightarrow X$ is a dominant rational self-map of a projective surface $X$ over $\mathbb{C}$ with a regular and non-invertible iterate $Φ^n$, then we can take $n \leq 12$. This bound is sharp and realized on $X = \mathbb{P}^2$. In the case where $Φ$ is a birational self-map of $\mathbb{P}^2$ we prove that as long as $Φ$ does not preserve a non-constant fibration, if some iterate $Φ^n$ is regular then $Φ$ itself must be regular.

math.AG↗
arXiv2021

On sparsity of representations of polynomials as linear combinations of exponential functions

Given an integer $g$ and also some given integers $m$ (sufficiently large) and $c_1,\dots, c_m$, we show that the number of all non-negative integers $n\le M$ with the property that there exist non-negative integers $k_1,\dots, k_m$ such that $$n^2=\sum_{i=1}^m c_i g^{k_i}$$ is $o\left(\left(\log M \right)^{m-1/2}\right)$. We also obtain a similar bound when dealing with more general inequalities $$\left|Q(n)-\sum_{i=1}^m c_iλ^{k_i}\right|\le B,$$ where $Q\in {\mathbb C}[X]$ and also $λ\in {\mathbb C}$ (while $B$ is a real number).

math.NT↗
arXiv2020

A sparsity result for the Dynamical Mordell-Lang Conjecture in positive characteristic

We prove a quantitative partial result in support of the Dynamical Mordell-Lang Conjecture (also known as the DML conjecture) in positive characteristic. More precisely, we show the following: given a field $K$ of characteristic $p$, given a semiabelian variety $X$ defined over a finite subfield of $K$ and endowed with a regular self-map $Φ:X \longrightarrow X$ defined over $K$, given a point $α\in X(K)$ and a subvariety $V\subseteq X$, then the set of all non-negative integers $n$ such that $Φ^n(α)\in V(K)$ is a union of finitely many arithmetic progressions along with a subset $S$ with the property that there exists a positive real number $A$ (depending only on $N$, $Φ$, $α$, $V$) such that for each positive integer $M$, we have $$\#\left\{n\in S\colon~ n\le M\right\}\le A\cdot \left(1+\log M\right)^{\dim V}.$$

math.NT↗