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Alina Ostafe

Publications and source records attributed to Alina Ostafe.

51 records · Page 3Linked to original sources
arXiv2017

Orbits of Polynomial Dynamical Systems Modulo Primes

We present lower bounds for the orbit length of reduction modulo primes of parametric polynomial dynamical systems defined over the integers, under a suitable hypothesis on its set of preperiodic points over $\mathbb C$. Applying recent results of Baker and DeMarco~(2011) and of Ghioca, Krieger, Nguyen and Ye~(2017), we obtain explicit families of parametric polynomials and initial points such that the reductions modulo primes have long orbits, for all but a finite number of values of the parameters. This generalizes a previous lower bound due to Chang~(2015). As a by-product, we also slighly improve a result of Silverman~(2008) and recover a result of Akbary and Ghioca~(2009) as special extreme cases of our estimates.

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arXiv2016

On roots of unity in orbits of rational functions

In this paper we characterise univariate rational functions over a number field $\K$ having infinitely many points in the cyclotomic closure $\K^c$ for which the orbit contains a root of unity. Our results are similar to previous results of Dvornicich and Zannier describing all polynomials having infinitely many preperiodic points in $\K^c$.

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arXiv2015

On the quantitative dynamical Mordell-Lang conjecture

The dynamical Mordell-Lang conjecture concerns the structure of the intersection of an orbit in an algebraic dynamical system and an algebraic variety. In this paper, we bound the size of this intersection for various cases when it is finite.

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arXiv2015

On some extensions of the Ailon-Rudnick Theorem

In this paper we present some extensions of the Ailon-Rudnick Theorem, which says that if $f,g\in{\mathbb C}[T]$, then $\gcd(f^n-1,g^m-1)$ is bounded for all $n,m\ge 1$. More precisely, using a uniform bound for the number of torsion points on curves and results on the intersection of curves with algebraic subgroups of codimension at least $2$, we present two such extensions in the univariate case. We also give two multivariate analogues of the Ailon-Rudnick Theorem based on Hilbert's irreducibility theorem and a result of Granville and Rudnick about torsion points on hypersurfaces.

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arXiv2014

Polynomial Values in Affine Subspaces of Finite Fields

In this paper we introduce a new approach and obtain new results for the problem of studying polynomial images of affine subspaces of finite fields. We improve and generalise several previous known results, and also extend the range of such results to polynomials of degrees higher than the characteristic of the field. Such results have a wide scope of applications similar to those associated with their counterparts studying consecutive intervals over prime fields instead of affine subspaces. Here we give only two immediate consequences: to a bound on the size of the intersection of orbits of polynomial dynamical systems with affine subspaces and to the Waring problem in affine subspaces. These results are based on estimates for a certain new type of exponential sums.

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arXiv2012

Stable Polynomials over Finite Fields

We use the theory of resultants of polynomials to study the stability of an arbitrary polynomial over a finite field, that is, the property of having all its iterates irreducible. This result partially generalises the quadratic polynomial case described by R. Jones and N. Boston. Moreover, for characteristic three, we show that certain polynomials of degree three are not stable. We also use the Weil bound for multiplicative character sums to estimate the number of stable arbitrary polynomials over finite fields of odd characteristic.

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arXiv2011

Degree Growth, Linear Independence and Periods of a Class of Rational Dynamical Systems

We introduce and study algebraic dynamical systems generated by triangular systems of rational functions. We obtain several results about the degree growth and linear independence of iterates as well as about possible lengths of trajectories generated by such dynamical systems over finite fields. Some of these results are generalisations of those known in the polynomial case, some are new even in this case.

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arXiv2011

On the Distribution of the Subset Sum Pseudorandom Number Generator on Elliptic Curves

Given a prime $p$, an elliptic curve $\E/\F_p$ over the finite field $\F_p$ of $p$ elements and a binary \lrs\ $\(u(n)\)_{n =1}^\infty$ of order~$r$, we study the distribution of the sequence of points $$ \sum_{j=0}^{r-1} u(n+j)P_j, \qquad n =1,..., N, $$ on average over all possible choices of $\F_p$-rational points $P_1,..., P_r$ on~$\E$. For a sufficiently large $N$ we improve and generalise a previous result in this direction due to E.~El~Mahassni.

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arXiv2010

Pseudorandomness and Dynamics of Fermat Quotients

We obtain some theoretic and experimental results concerning various properties (the number of fixed points, image distribution, cycle lengths) of the dynamical system naturally associated with Fermat quotients acting on the set $\{0, ..., p-1\}$. We also consider pseudorandom properties of Fermat quotients such as joint distribution and linear complexity.

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arXiv2010

Some Additive Combinatorics Problems in Matrix Rings

We study the distribution of singular and unimodular matrices in sumsets in matrix rings over finite fields. We apply these results to estimate the largest prime divisor of the determinants in sumsets in matrix rings over the integers.

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arXiv2010

Multivariate Permutation Polynomial Systems and Nonlinear Pseudorandom Number Generators

In this paper we study a class of dynamical systems generated by iterations of multivariate permutation polynomial systems which lead to polynomial growth of the degrees of these iterations. Using these estimates and the same techniques studied previously for inversive generators, we bound exponential sums along the orbits of these dynamical systems and show that they admit much stronger estimates on average over all initial values than in the general case and thus can be of use for pseudorandom number generation.

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arXiv2010

Pseudorandom Numbers and Hash Functions from Iterations of Multivariate Polynomials

Dynamical systems generated by iterations of multivariate polynomials with slow degree growth have proved to admit good estimates of exponential sums along their orbits which in turn lead to rather stronger bounds on the discrepancy for pseudorandom vectors generated by these iterations. Here we add new arguments to our original approach and also extend some of our recent constructions and results to more general orbits of polynomial iterations which may involve distinct polynomials as well. Using this construction we design a new class of hash functions from iterations of polynomials and use our estimates to motivate their "mixing" properties.

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arXiv2009

On the Length of Critical Orbits of Stable Quadratic Polynomials

We use the Weil bound of multiplicative character sums together with some recent results of N. Boston and R. Jones, to show that the critical orbit of quadratic polynomials over a finite field of $q$ elements is of length $O(q^{3/4})$, improving upon the trivial bound $q$.

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arXiv2009

On the Degree Growth in Some Polynomial Dynamical Systems and Nonlinear Pseudorandom Number Generators

In this paper we study a class of dynamical systems generated by iterations of multivariate polynomials and estimate the degreegrowth of these iterations. We use these estimates to bound exponential sums along the orbits of these dynamical systems and show that they admit much stronger estimates than in the general case and thus can be of use for pseudorandom number generation.

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