Related papers: The Newman algorithm for constructing polynomials …

Analysing the cubic sectors of a real polynomial of degree n, a modification of the Newton Rule is Signs is proposed with which stricter upper bound on the number of real roots can be found. A new necessary condition for reality of the…

A regularity lemma for polynomials provides a decomposition in terms of a bounded number of approximately independent polynomials. Such regularity lemmas play an important role in numerous results, yet suffer from the familiar shortcoming…

We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials $H_n(z):=\sum_{j=1}^{m_n} a_jp_j(z)$ that are linear combinations of basis polynomials $\{p_j\}$…

It is shown that any smooth closed orientable manifold of dimension $2k + 1$, $k \geq 2$, admits a smooth polynomially convex embedding into $\mathbb C^{3k}$. This improves by $1$ the previously known lower bound of $3k+1$ on the possible…

We show how to construct sparse polynomial systems that have non-trivial lower bounds on their numbers of real solutions. These are unmixed systems associated to certain polytopes. For the order polytope of a poset P this lower bound is the…

The presented analysis determines several new bounds on the roots of the equation $a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 = 0$ (with $a_n > 0$). All proposed new bounds are lower than the Cauchy bound max$\{1, \sum_{j=0}^{n-1} |a_j/a_n|…

We generalize Sylvester single sums to multisets (sets with repeated elements), and show that these sums compute subresultants of two univariate polyomials as a function of their roots independently of their multiplicity structure. This is…

Consider a polynomial vector field $\xi$ in $\mathbb{C}^n$ with algebraic coefficients, and $K$ a compact piece of a trajectory. Let $N(K,d)$ denote the maximal number of isolated intersections between $K$ and an algebraic hypersurface of…

We develop a new tool, namely polynomial and linear algebraic methods, for studying systems of word equations. We illustrate its usefulness by giving essentially simpler proofs of several hard problems. At the same time we prove extensions…

We study the regularity of the roots of multiparameter families of complex univariate monic polynomials $P(x)(z) = z^n + \sum_{j=1}^n (-1)^j a_j(x) z^{n-j}$ with fixed degree $n$ whose coefficients belong to a certain subring $\mathcal C$…

Consider $\{p_n\}_{n=0}^{\infty}$, a sequence of polynomials orthogonal with respect to $w(x)>0$ on $(a,b)$, and polynomials $\{g_{n,k}\}_{n=0}^{\infty},k \in \mathbb{N}_0$, orthogonal with respect to $c_k(x)w(x)>0$ on $(a,b)$, where…

Given a natural number $n \geq 4$ we show that there exists infinitely many polynomials $f_{n}(x):= \prod_{i=1}^{n} (x^{2} - a_{i})$ such that (i) $f_{n}(x)$ has a root modulo every positive integer, (ii) $f_{n}(x)$ has no rational roots,…

Given a linear recurrence of the form $c_n=a_1c_{n-1}+\cdots+a_j c_{n-j}$, it is well-known that $c_n=\sum_{r}p_r(n)r^n$, where the sum is taken over the set of characteristic roots and each $p_r(n)$ is some polynomial. We give a closed…

We present a combination of two algorithms that accurately calculate multiple roots of general polynomials. Algorithm I transforms the singular root-finding into a regular nonlinear least squares problem on a pejorative manifold, and…

In this paper we shall find some sufficient conditions for a uniqueness polynomial to be a strong uniqueness polynomial, as this type of problem was never investigated by researchers earlier.We also exhibit some examples to substantiate our…

We opt to study the convergence of maximal real roots of certain Fibonacci-type polynomials given by $G_n=x^kG_{n-1}+G_{n-2}$. The special cases $k=1$ and $k=2$ are found in [4] and [7], respectively.

We begin with the observation that the signed generalized Stirling polynomials $P_k(m,x)$, which occur in a generalization of Malmsten's integral, reduce to the falling factorials when $k=m$. The structure of these generalized Stirling…

Newton's method for polynomial root finding is one of mathematics' most well-known algorithms. The method also has its shortcomings: it is undefined at critical points, it could exhibit chaotic behavior and is only guaranteed to converge…

Let $\{B_n\}$, $\{B_n(x)\}$ and $\{E_n(x)\}$ be the Bernoulli numbers, Bernoulli polynomials and Euler polynomials, respectively. In this paper we mainly establish formulas for $\sum_{6\mid k-3}\binom nkB_{n-k}(x)$, $\sum_{6\mid k}\binom…

In this paper, we investigate condition numbers of eigenvalue problems of matrix polynomials with nonsingular leading coefficients, generalizing classical results of matrix perturbation theory. We provide a relation between the condition…