Related papers: Random complex fewnomials, I
This paper considers the approximation of a monomial $x^n$ over the interval $[-1,1]$ by a lower-degree polynomial. This polynomial approximation can be easily computed analytically and is obtained by truncating the analytical Chebyshev…
Let $S$ be a rational fraction and let $f$ be a polynomial over a finite field. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. In certain cases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. For instance,…
We introduce a finite version of free probability for rectangular matrices that amounts to operations on singular values of polynomials. We show that we can replicate the transforms from free probability, and that asymptotically there is…
We show that for Gaussian random SU(m+1) polynomials of a large degree N the probability that there are no zeros in the disk of radius r is less than $e^{-c_{1,r} N^{m+1}}$, and is also greater than $e^{-c_{2,r} N^{m+1}}$. Enroute to this…
We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This…
We study the regularity of densities of distributions that are polynomial images of the standard Gaussian measure on $\mathbb{R}^n$. We assume that the degree of a polynomial is fixed and that each variable enters to a power bounded by…
In this article, we consider the singularity of an arbitrary homogeneous polynomial with complex coefficients $f(x_0,\dots,x_n)$ at the origin of $\mathbb C^{n+1}$, via the study of the monodromy characteristic polynomials $\Delta_l(t)$,…
Consider a random system $\mathfrak{f}_1(x)=0,\ldots,\mathfrak{f}_n(x)=0$ of $n$ random real polynomials in $n$ variables, where each $\mathfrak{f}_k$ has a prescribed set of exponent vectors in a set $A_k\subseteq \mathbb{Z}^n$ of size…
In this work we relate the deterministic complexity of factoring polynomials (over finite fields) to certain combinatorial objects we call m-schemes. We extend the known conditional deterministic subexponential time polynomial factoring…
For a fixed prime power $q$ and natural number $d$ we consider a random polynomial $$f=x^n+a_{n-1}(t)x^{n-1}+\ldots+a_1(t)x+a_0(t)\in\mathbb F_q[t][x]$$ with $a_i$ drawn uniformly and independently at random from the set of all polynomials…
This paper introduces a novel class of prime-generating quadratic polynomials defined by $f_{Z,k,H}(n) = n^2 - (2Zk - 1)n + \frac{(2Zk - 1)^2 + H}{4}$, where $Zk \in \mathbb{Z}_{\geq 0}$ and $H$ belongs to the set of Heegner numbers. This…
We prove that the number of distinct homotopy types of limits of one-parameter semi-algebraic families of closed and bounded semi-algebraic sets is bounded singly exponentially in the additive complexity of any quantifier-free first order…
This paper is our second step towards developing a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by an arithmetic circuit has some types of monomials in its…
We study divisibility properties of a set $\{f_1(\mathbf{U}_n^{(s)}),\ldots,f_m(\mathbf{U}_n^{(s)})\}$, where $f_1,\ldots,f_m$ are polynomials in $s$ variables over $\mathbb{Z}$ and $\mathbf{U}_n^{(s)}$ is a point picked uniformly at random…
In this paper, we present a probabilistic extension of the Fubini polynomials and numbers associated with a random variable satisfying some appropriate moment conditions. We obtain the exponential generating function and an integral…
A result of Hoskins and Steinerberger [Int. Math. Res. Not., (13):9784-9809, 2022] states that repeatedly differentiating a random polynomials with independent and identically distributed mean zero and variance one roots will result, after…
Binomial ideals are special polynomial ideals with many algorithmically and theoretically nice properties. We discuss the problem of deciding if a given polynomial ideal is binomial. While the methods are general, our main motivation and…
The Bohnenblust-Hille inequality and its variants have found applications in several areas of Mathematics and related fields. The control of the constants for the variant for complex $m$-homogeneous polynomials is of special interest for…
Suppose $F:=(f_1,\ldots,f_n)$ is a system of random $n$-variate polynomials with $f_i$ having degree $\leq\!d_i$ and the coefficient of $x^{a_1}_1\cdots x^{a_n}_n$ in $f_i$ being an independent complex Gaussian of mean $0$ and variance…
Consider a polynomial of large degree n whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly k real zeros…