Related papers: On asymptotic Gauss-Lucas theorem

The main results of this article are asymptotic formulas for the variance of the number of zeros of a Gaussian random polynomial of degree $N$ in an open set $U \subset C$ as the degree $N \to \infty$, and more generally for the zeros of…

Let $\mu$ be a probability measure on $\mathbb C$, and let $P_n$ be the random polynomial whose zeros are sampled independently from $\mu$. We study the asymptotic distribution of zeros of high-order derivatives of $P_n$. We show that, for…

We develop a Galois theory of commutative rings under actions of finite inverse semigroups. We present equivalences for the definition of Galois extension as well as a Galois correspondence theorem. We also show how the theory behaves in…

We find an inverse factorial series expansion for the ratio of products of gamma functions whose arguments are linear functions of the variable. We a give recurrence relation for the coefficients in terms of the N{\o}rlund-Bernoulli…

Recently Rather et al. \cite{NT} considered the generalized derivative and the generalized polar derivative and studied the relative position of zeros of generalized derivative and generalized polar derivative with respect to the zeros of…

We prove that the $abc$-Conjecture implies upper bounds on Zsigmondy sets that are uniform over families of unicritical polynomials over number fields. As an application, we use the $abc$-Conjecture to prove that there exist uniform bounds…

We investigate the probability that a random polynomial with independent, mean-zero and finite variance coefficients has no real zeros. Specifically, we consider a random polynomial of degree $2n$ with coefficients given by an i.i.d.…

We present an encoding of a polynomial system into vanishing and non-vanishing constraints on almost-principal minors of a symmetric, principally regular matrix, such that the solvability of the system over some field is equivalent to the…

An approach is proposed for bounding the number of zeros that solutions of linear differential systems with polynomial coefficients may have. A bound is obtained in a special case which improves upon currently existing.

We give abstract versions of the large deviation theorem for the distribution of zeros of polynomials and apply them to the characteristic polynomials of Hermitian random matrices. We obtain new estimates related to the local semi-circular…

We prove that, for asymptotically bounded holomorphic functions in a sector in $\mathbb{C}$, an asymptotic expansion in a single direction towards the vertex with constraints in terms of a logarithmically convex sequence admitting a nonzero…

The classic Gauss-Lucas Theorem for complex polynomials of degree $d\ge2$ has a natural reformulation over quaternions, obtained via rotation around the real axis. We prove that such a reformulation is true only for $d=2$. We present a new…

A partition polynomial is a refinement of the partition number p(n) whose coefficients count some special partition statistic. Just as partition numbers have useful asymptotics so do partition polynomials. In fact, their asymptotics…

We study equidistribution problem of zeros in relation to a sequence of $Z$-asymptotically Chebyshev polynomials on $\mathbb{C}^{m}$. We use certain results obtained in a very recent work of Bayraktar, Bloom and Levenberg and have an…

We estimate the number of zeros of a polynomial in $\mathbb{C}[z]$ within any small circular disc centered on the unit circle, which improves and comprehensively extends a result established by Borwein, Erd{\'e}lyi, and Littmann~\cite{BE1}…

Let $W_X(\mathbb{F})$ be the Lie algebra of all derivations of the polynomial algebra $\mathbb{F}[X]$ in infinitely many variables. We describe all derivations of $W_X(\mathbb{F})$ over a field of characteristic zero and prove that all such…

According to the generalized Polya theorem, the Gaussian distribution on the real line is characterized by the property of equidistribution of a monomial and a linear form of independent identically distributed random variables. We give a…

We give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well-known families of polynomials, and the interplay between these models and the asymptotic distribution of their…

We study distribution of zeros of a complex polynomial whose coefficients has been modified. We give a new proof of the theorem of Rubinstein, and with similar method we prove a new theorem that is not generalization of the previous…

We consider a certain class of multiplicative functions $f: \mathbb N \rightarrow \mathbb C$ and study the distribution of zeros of Dirichlet polynomials $F_N(s)= \sum_{n\le N} f(n)n^{-s}$ corresponding to these functions. We prove that the…