Related papers: Discriminant and root separation of integral polyn…

We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in $B_{\alpha} \in L[y]$, where $L=\QQ(\alpha)$ is a simple algebraic extension of the rational numbers. We…

We consider the Rosenfeld-Groebner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials…

Let $\mu$ and $\nu$ be probability measures in the complex plane, and let $p$ and $q$ be independent random polynomials of degree $n$, whose roots are chosen independently from $\mu$ and $\nu$, respectively. Under assumptions on the…

Let $k \geq 2$ be an integer and $\mathbb F_q$ be a finite field with $q$ elements. We prove several results on the distribution in short intervals of polynomials in $\mathbb F_q[x]$ that are not divisible by the $k$th power of any…

We study the number of prime polynomials of degree $n$ over $\mathbb{F}_q$ in which the $i^{th}$ coefficient is either preassigned to be $a_i \in \mathbb{F}_q$ or outside a small set $S_i \subset \mathbb{F}_q$. This serves as a function…

We consider a class of real random polynomials, indexed by an integer d, of large degree n and focus on the number of real roots of such random polynomials. The probability that such polynomials have no real root in the interval [0,1]…

The purpose of this note is to characterize those orthogonal polynomials sequences $(P_n)_{n\geq0}$ for which $$ \pi(x)\mathcal{D}_q P_n(x)=(a_n x+b_n)P_n(x)+c_n P_{n-1}(x),\quad n=0,1,2,\dots, $$ where $\mathcal{D}_q$ is the Askey-Wilson…

We prove some uniqueness results which improve and generalize results of Jiang-Tao Li and Ping Li[Uniqueness of entire functions concerning differential polynomials. Commun. Korean Math. Soc. 30 (2015), No. 2, pp. 93-101].

Let $(G,\rho)$ be a stationary random graph, and use $B^G_{\rho}(r)$ to denote the ball of radius $r$ about $\rho$ in $G$. Suppose that $(G,\rho)$ has annealed polynomial growth, in the sense that $\mathbb{E}[|B^G_{\rho}(r)|] \leq O(r^k)$…

In this paper, we tackle the following problem: compute the gcd for several univariate polynomials with parametric coefficients. It amounts to partitioning the parameter space into ``cells'' so that the gcd has a uniform expression over…

We consider the following basic, and very broad, statistical problem: Given a known high-dimensional distribution ${\cal D}$ over $\mathbb{R}^n$ and a collection of data points in $\mathbb{R}^n$, distinguish between the two possibilities…

A univariate polynomial f over a field is decomposable if f = g o h = g(h) for nonlinear polynomials g and h. It is intuitively clear that the decomposable polynomials form a small minority among all polynomials over a finite field. The…

Let f(x) = f(x_1, ..., x_n) = \sum_{|S| <= k} a_S \prod_{i \in S} x_i be an n-variate real multilinear polynomial of degree at most k, where S \subseteq [n] = {1, 2, ..., n}. For its "one-block decoupled" version, f~(y,z) = \sum_{|S| <= k}…

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a real coefficient polynomial. They can be approximated at a low computational cost if the…

It is proved that the universal degree bound for separating polynomial invariants of a finite abelian group (in non-modular characteristic) is strictly smaller than the universal degree bound for generators of polynomial invariants, unless…

We consider the distribution of spacings between consecutive elements in subsets of Z/qZ where q is highly composite and the subsets are defined via the Chinese remainder theorem. We give a sufficient criterion for the spacing distribution…

If $f$ is a polynomial with all of its roots on the real line, then the roots of the derivative $f'$ are more evenly spaced than the roots of $f$. The same holds for a real entire function of order~1 with all its zeros on a line. In…

Let $N$ be a positive integer and let $S_N$ be the set of polynomials with integer coefficients, degree less than $N$, and minimal positive integral over $[0,1]$. D. Bazzanella initiated the study of $S_N$ because of its relation to the…

In this paper, we study distributional properties of the sequence of partial quotients in the continued fraction expansion of fractions $a/N$, where $N$ is fixed and $a$ runs through the set of mod $N$ residue classes which are coprime with…

Let $1<t<n$ be integers, where $t$ is a divisor of $n$. An R-$q^t$-partially scattered polynomial is a $\mathbb F_q$-linearized polynomial $f$ in $\mathbb F_{q^n}[X]$ that satisfies the condition that for all $x,y\in\mathbb F_{q^n}^*$ such…