Related papers: Seeking a quadratic refinement of Sendov's conject…

We consider the class of all homogeneous, possibly non-reduced, polynomials $f$ whose associated reduced projective divisor $D_{\text{red}} \subset \mathbb{P}^{n-1}$ has (at worst) quasi-homogeneous isolated singularities. In an arbitrary…

A polynomial $p\in\mathbb{R}[x]$ is a divisor of some polynomial $0\neq f\in\mathbb{R}[x]$ with non-negative coefficients if and only if $p$ does not have a positive real root. The lowest possible degree of such $f$ for a given $p$ is known…

We give an upper bound in O(d ^((n+1)/2)) for the number of critical points of a normal random polynomial with degree d and at most n variables. Using the large deviation principle for the spectral value of large random matrices we obtain…

Theorem 1 is a formula expressing the mean number of real roots of a random multihomogeneous system of polynomial equations as a multiple of the mean absolute value of the determinant of a random matrix. Theorem 2 derives closed form…

Given a polynomial $p$, the degree of its Chebyshev's method $C_p$ is determined. If $p$ is cubic then the degree of $C_p$ is found to be $4,6$ or $7$ and we investigate the dynamics of $C_p$ in these cases. If a cubic polynomial $p$ is…

Let p(z) be a complex polynomial of degree n. Let C be a circle containing its n-1 zeros, having its center in the centroid of these zeros. We prove that C must contain at least int((n-1):2) zeros of its derivative.

In this paper, we obtain new results on the critical points of a polynomial, these results are useful to the Sendov conjecture.

In this paper, we consider the relationship between the Mahler measure of a polynomial and its separation. In 1964, Mahler proved that if $f(x) \in \mathbb{Z}[x]$ is separable of degree $n$, then $\operatorname{sep}(f) \gg_n M(f)^{-(n-1)}$.…

Let $d\geq 2$ and $k\geq 1$ be fixed. We prove that, for every $\epsilon>0$ and every real $\beta$, there exist integers $1\leq b_1,\ldots,b_k\leq N$ such that \[ \left\|\sum_{j=1}^k b_j^{1/d}-\beta\right\| \ll_{d,k,\epsilon}…

We study the regularity of the roots of complex univariate polynomials whose coefficients depend smoothly on parameters. We show that any continuous choice of the roots of a $C^{n-1,1}$-curve of monic polynomials of degree $n$ is locally…

Consider a random polynomial $$ G_Q(x)=\xi_{Q,n}x^n+\xi_{Q,n-1}x^{n-1}+...+\xi_{Q,0} $$ with independent coefficients uniformly distributed on $2Q+1$ integer points $\{-Q, ..., Q\}$. Denote by $D(G_Q)$ the discriminant of $G_Q$. We show…

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 +…

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…

Let $\mathcal{E}_d^{(s)}$ denote the set of coefficient vectors $(a_1,\dots,a_d)\in \mathbb{R}^d$ of contractive polynomials $x^d+a_1x^{d-1}+\dots+a_d\in \mathbb{R}[x]$ that have exactly $s$ pairs of complex conjugate roots and let…

We use Newton's method to find all roots of several polynomials in one complex variable of degree up to and exceeding one million and show that the method, applied to appropriately chosen starting points, can be turned into an algorithm…

We prove that for every $\varepsilon>0$ and a nonnegative integer $\omega$ there exist primes $p_1,p_2,\ldots,p_\omega$ such that for $n=p_1p_2\ldots p_\omega$ the height of the cyclotomic polynomial $\Phi_n$ is at least…

Several conditions are known for a self-inversive polynomial that ascertain the location of its roots, and we present a framework for comparison of those conditions. We associate a parametric family of polynomials $p_\alpha$ to each such…

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…

The Casas-Alvero conjecture says that a degree $n$ complex univariate polynomial sharing a root with each of its derivative must have only one root. In this article we give three results. The first one, is that the number of possible…

The Casas-Alvero conjecture states: if a complex univariate polynomial has a common root with each of its derivatives, then it has a unique root. We show that hypothetical counterexamples must have at least 5 different roots. The first case…