Related papers: The Newman algorithm for constructing polynomials …
Polynomials with all the coefficients in $\{ 0,1\}$ and constant term 1 are called Newman polynomials, whereas polynomials with all the coefficients in $\{ -1,1\}$ are called Littlewood polynomials. By exploiting an algorithm developed…
For each $\alpha \in (0, 1)$, we construct a bounded monotone deterministic sequence $(c_k)_{k \geq 0}$ of real numbers so that the number of real roots of the random polynomial $f_n(z) = \sum_{k=0}^n c_k \varepsilon_k z^k$ is $n^{\alpha +…
We consider the problem of finding a condition for a univariate polynomial having a given multiplicity structure when the number of distinct roots is given. It is well known that such conditions can be written as conjunctions of several…
A Newman polynomial has all the coefficients in $\{ 0,1\}$ and constant term 1, whereas a Littlewood polynomial has all coefficients in $\{-1,1\}$. We call $P(X)\in\mathbb{Z}[X]$ a Borwein polynomial if all its coefficients belong to $\{…
We consider random polynomials $p_n(x)=\xi_0+\xi_1+\dots+\xi_n x^n$ whose coefficients are independent and identically distributed with zero mean, unit variance, and bounded $(2+\epsilon)^{th}$ moment (for some $\epsilon>0$), also known as…
In this paper, we extend the classical Newton-Maclaurin inequalities to functions $S_{k;s}(x)=E_k(x)+\dsum_{i=1}^s \al_i E_{k-i}(x)$, which are formed by linear combinations of multiple basic symmetric mean. We proved that when the…
In this article, we construct explicit examples of pairs of non-isomorphic trees with the same restricted $U$-polynomial for every $k$; by this we mean that the polynomials agree on terms with degree at most $k+1$. The main tool for this…
Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…
The analysis of many physical phenomena can be reduced to the study of solutions of differential equations with polynomial coefficients. In the present work, we establish the necessary and sufficient conditions for the existence of…
Multiplicative relations between the roots of a polynomial in $\mathbb{Q}[x]$ have drawn much attention in the field of arithmetic and algebra, while the problem of computing these relations is interesting to researchers in many other…
We study $\{0, 1\}$ and $\{-1, 1\}$ polynomials $f(z)$, called Newman and Littlewood polynomials, that have a prescribed number $N(f)$ of zeros in the open unit disk $\mathcal{D} = \{z \in \mathbb{C}: |z| < 1\}$. For every pair $(k, n) \in…
In this paper, we prove that if $f(x)=\sum_{k=0}^n{n\choose k}a_kx^k$ is a polynomial with real zeros only, then the sequence $\{a_k\}_{k=0}^n$ satisfies the following inequalities $a_{k+1}^2(1-\sqrt{1-c_k})^2/a_k^2…
In this paper, we study the root distribution of some univariate polynomials $W_n(z)$ satisfying a recurrence of order two with linear polynomial coefficients over positive numbers. We discover a sufficient and necessary condition for the…
We state a kind of Euclidian division theorem: given a polynomial P(x) and a divisor d of the degree of P, there exist polynomials h(x),Q(x),R(x) such that P(x) = h(Q(x)) +R(x), with deg h=d. Under some conditions h,Q,R are unique, and Q is…
Given n polynomials in n variables of respective degrees d_1,...,d_n, and a set of monomials of cardinality d_1...d_n, we give an explicit subresultant-based polynomial expression in the coefficients of the input polynomials whose…
We show that smooth curves of monic complex polynomials $P_a (Z)=Z^n+\sum_{j=1}^n a_j Z^{n-j}$, $a_j : I \to \mathbb C$ with $I \subset \mathbb R$ a compact interval, have absolutely continuous roots in a uniform way. More precisely, there…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…
Given a natural number $n \geq 2$, an integer $k$ and for a judiciously chosen $l = l(n)$ we give necessary and sufficient conditions for the polynomial $f_{n,k} = \big( \sum_{i=1}^{l} x_{i}^{n} \big) - k$ to have roots modulo every…
In this note we characterize when non-classical polynomials are necessary in the inverse theorem for the Gowers $U^k$-norm. We give a brief deduction of the fact that a bounded function on $\mathbb F_p^n$ with large $U^k$-norm must…
Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of…