Alina Ostafe

Systems of equations with elements of finitely generated groups and with linear recurrence sequences

Subham Bhakta, Alina Ostafe, Igor E. Shparlinski

We obtain new and essentially optimal bounds on the number of solutions of diagonal systems of $k$ polynomial equations in $s$ variables from an arbitrary finite subset of a finite…

Powers in orbits and semigroups generated by polynomials

Alina Ostafe, Dat T. Tran

In this paper, we study the presence of perfect powers in orbits of polynomials, or of semigroups generated by a finite family of polynomials, defined over a number field. In parti…

The divisor function for matrices

Tim Browning, Nikita P. Kalinin, Alina Ostafe +2

We introduce a matrix divisor function $τ_{n} (T, M)$ , counting factorisations $A B = M$ for $n \times n$ integer matrices $A, B$ of height at most $T$ . For a fixed non-singular $M$ , or for…

Counting Irreducible polynomials with coefficients from thin subgroups

Alina Ostafe, Igor E. Shparlinski

L. Bary-Soroker and R. Shmueli (2026) have given an asymptotic formula for the number of irreducible polynomials over the finite fields $F_{q}$ of $q$ elements, such that the…

Multiplicative dependence in the denominators of points of elliptic curves

Attila Bérczes, Subham Bhakta, Lajos Hajdu +2

Let $E_{1}, \dots, E_{s}$ be $s$ , not necessary distinct, elliptic curves over $Q$ . We give upper bounds on the frequency of $s$ -tuples of points in $E_1(\mathbb{Q})\times \…

Counting solvable $S$ -unit equations and linear recurrence sequences with zeros

Alina Ostafe, Carl Pomerance, Igor E. Shparlinski

We show that only a rather small proportion of linear equations are solvable in elements of a fixed finitely generated subgroup of a multiplicative group of a number field. The arg…

Counting integer matrices with square-free determinants

Alina Ostafe, Igor E. Shparlinski

We consider the set $M_{n} (Z; H)$ of $n \times n$ -matrices with integer elements of size at most $H$ and obtain an asymptotic formula for the number of…

Counting matrices over finite rank multiplicative groups

Aaron Manning, Alina Ostafe, Igor E. Shparlinski

Motivated by recent works on statistics of matrices over sets of number theoretic interest, we study matrices with entries from arbitrary finite subsets $A$ of finite rank…

math.NT2025★ 1 cited

Multiplicative dependence in linear recurrence sequences

Attila Bérczes, Lajos Hajdu, Alina Ostafe +1

For a wide class of integer linear recurrence sequences $(u (n))_{n = 1}^{\infty}$ , we give an upper bound on the number of $s$ -tuples $\left(n_1, \ldots, n_s\right) \in \lef…

math.DS2024

On quantum ergodicity for higher dimensional cat maps

Pär Kurlberg, Alina Ostafe, Zeev Rudnick +1

We study eigenfunction localization for higher dimensional cat maps, a popular model of quantum chaos. These maps are given by linear symplectic maps in ${\mathrm{Sp}}(2g,\mathbb Z…

math.NT2024

On the frequency of primes preserving dynamical irreducibility of polynomials

Alina Ostafe, Igor E. Shparlinski

Towards a well-known open question in arithmetic dynamics, L. Mérai, A. Ostafe and I. E. Shparlinski (2023), have shown, for a class of polynomials $f \in Z [X]$ , which in p…

math.NT2024

Statistics of ranks, determinants and characteristic polynomials of rational matrices

Muhammad Afifurrahman, Vivian Kuperberg, Alina Ostafe +1

We consider the set of $n \times n$ matrices with rational entries having numerator and denominator of size at most $H$ and obtain upper and lower bounds on the number of such matri…

math.NT2023

On the sparsity of non-diagonalisable integer matrices and matrices with a given discriminant

Alina Ostafe, Igor E. Shparlinski

We consider the set $M_{n} (Z; H)$ of $n \times n$ -matrices with integer elements of size at most $H$ and obtain upper bounds on the number of matrices from $\mathcal…

math.NT2023

On some matrix counting problems

Ali Mohammadi, Alina Ostafe, Igor Shparlinski

We estimate the frequency of singular matrices and of matrices of a given rank whose entries are parametrised by arbitrary polynomials over the integers and modulo a prime $p$ . In…

math.NT2023

Explicit bounds for the solutions of superelliptic equations over number fields

Attila Bérczes, Yann Bugeaud, Kálmán Győry +3

Let $f$ be a polynomial with coefficients in the ring $O_{S}$ of $S$ -integers of a number field $K$ , $b$ a non-zero $S$ -integer, and $m$ an integer $\geq 2$ . We consider the equation…

math.NT2023★ 1 cited

Counting embeddings of free groups into $SL_{2} (Z)$ and its subgroups

Kamil Bulinski, Alina Ostafe, Igor E. Shparlinski

We show that if one selects uniformly independently and identically distributed matrices $A_{1}, \dots, A_{s} \in SL_{2} (Z)$ from a ball of large radius $X$ then with…

math.NT2022

Weil Sums over Small Subgroups

Alina Ostafe, Igor E. Shparlinski, José Felipe Voloch

We obtain new bounds on short Weil sums over small multiplicative subgroups of prime finite fields which remain nontrivial in the range the classical Weil bound is already trivial.…

math.NT2022

Cyclotomic and abelian points in backward orbits of rational functions

Andrea Ferraguti, Alina Ostafe, Umberto Zannier

We prove several results on backward orbits of rational functions over number fields. First, we show that if $K$ is a number field, $ϕ \in K (x)$ and $α \in K$ then the extension of $…

math.NT2022★ 1 cited

Integer matrices with a given characteristic polynomial and multiplicative dependence of matrices

Philipp Habegger, Alina Ostafe, Igor E. Shparlinski

We consider the set $M_{n} (Z; H)$ of $n \times n$ -matrices with integer elements of size at most $H$ and obtain a new upper bound on the number of matrices from $\ma…

Equations and character sums with matrix powers, Kloosterman sums over small subgroups and quantum ergodicity

Alina Ostafe, Igor E. Shparlinski, José Felipe Voloch

We obtain a nontrivial bound on the number of solutions to the equation $A^{x_{1}} + \dots + A^{x_{ν}} = A^{x_{ν + 1}} + \dots + A^{x_{2 ν}}, 1 \leq x_{1}, \dots, x_{2 ν} \leq τ,$ …

Additive energy of cyclic matrix groups and character sums with matrix exponential functions

Alina Ostafe, Igor E. Shparlinski

We obtain a nontrivial bound on the number of solutions to the equation $A^{x_{1}} + A^{x_{2}} = A^{x_{3}} + A^{x_{4}}$ , $1 \leq x_{1}, x_{2}, x_{3}, x_{4} \leq τ$ , with a fixed $n \times n$ matrix $A$ …

math.NT2021★ 1 cited

Multiplicative dependence of rational values modulo approximate finitely generated groups

Attila Bérczes, Yann Bugeaud, Kálmán Győry +3

In this paper, we establish some finiteness results about the multiplicative dependence of rational values modulo sets which are `close' (with respect to the Weil height) to divisi…

On a Problem of Lang for Matrix Polynomials

Alina Ostafe

In this paper, we consider a problem of Lang about finiteness of torsion points on plane rational curves over $C$ , and prove some results towards a matrix analogue of this…