Related papers: Free infinite divisibility, fractional convolution…

We prove an analogue of the classical Bateman-Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable $x$)…

In this paper we study Appell polynomials by connecting them to random variables. This probabilistic approach yields, e.g., the mean value property which is fundamental in the sense that many other properties can be derived from it. We also…

Let $p$ be a prime number and let $S=\{x^p+c_1,\dots,x^p+c_r\}$ be a finite set of unicritical polynomials for some $c_1,\dots,c_r\in\mathbb{Z}$. Moreover, assume that $S$ contains at least one irreducible polynomial over $\mathbb{Q}$. Then…

The notion of root polynomials of a polynomial matrix $P(\lambda)$ was thoroughly studied in [F. Dopico and V. Noferini, Root polynomials and their role in the theory of matrix polynomials, Linear Algebra Appl. 584:37--78, 2020]. In this…

We develop analytic tools for studying the free multiplicative convolution of any measure on the real line and any measure on the nonnegative real line. More precisely, we construct the subordination functions and the $S$-transform of an…

We investigate the evolution of the empirical distribution of the complex roots of high-degree random polynomials, when the polynomial undergoes the heat flow. In one prominent example of Weyl polynomials, the limiting zero distribution…

We introduce a canonical notion of entropy for polynomials analogue to that of random variables in probability. We prove that entropy increases smoothly with respect to finite free addition. In particular we get the new inequality : $…

We define an extension of the polynomial calculus on a W*-probability space by introducing an abstract algebra which contains polynomials. This extension allows us to define transition operators for additive and multiplicative free…

Since the celebrated resolution of Kadison-Singer (via the Paving Conjecture) by Marcus, Spielman, and Srivastava, much study has been devoted to further understanding and generalizing the techniques of their proof. Specifically, their…

For a subgroup of $PGL(2,q)$ we show how some irreducible polynomials over $\mathbb{F}_q$ arise from the field of invariant rational functions. The proofs rely on two actions of $PGL(2,F)$, one on the projective line over a field $F$ and…

The framework of algebraically natural proofs was independently introduced in the works of Forbes, Shpilka and Volk (2018), and Grochow, Kumar, Saks and Saraf (2017), to study the efficacy of commonly used techniques for proving lower…

Debbah and Ryan have recently proved a result about the limit empirical singular distribution of the sum of two rectangular random matrices whose dimensions tend to infinity. In this paper, we reformulate it in terms of the rectangular free…

This paper is an exposition, with some new applications, of our results on the growth of entropy of convolutions. We explain the main result on $\mathbb{R}$, and derive, via a linearization argument, an analogous result for the action of…

One of the main applications of free probability is to show that for appropriately chosen independent copies of $d$ random matrix models, any noncommutative polynomial in these $d$ variables has a spectral distribution that converges…

In this article we introduce powerful tools and techniques from invariant theory to free analysis. This enables us to study free maps with involution. These maps are free noncommutative analogs of real analytic functions of several…

We investigate the asymptotic behavior of the eigenvalues of spiked perturbations of Wigner matrices when the dimension goes to infinity. The entries of the Hermitian Wigner matrix have a distribution which is symmetric and satisfies a…

In [1] Peters and Regts confirmed a conjecture by Sokal by showing that for every $\Delta \in \mathbb{Z}_{\geq 3}$ there exists a complex neighborhood of the interval $\left[0, \frac{\left(\Delta - 1\right)^{\Delta -…

We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model. Our first result shows…

We show that the counts of low degree irreducible factors of a random polynomial $f$ over $\mathbb{F}_q$ with independent but non-uniform coefficients behave like that of a uniform random polynomial, exhibiting a form of universality for…

A 1993 result of Alon and F\"uredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to…