Related papers: Inradius of random lemniscates

This article presents some interesting and novel results concerning the average modulus of random polynomials on the unit circle and the unit disc, with coefficients distributed as standard normal variates. The paper also introduces new…

We study the roots of a random polynomial over the field of p-adic numbers. For a random monic polynomial with coefficients in $\mathbb{Z}_p$, we obtain an asymptotic formula for the factorial moments of the number of roots of this…

If A_1,...,A_N are real square matrices then the p-radius, generalised Lyapunov exponent or matrix pressure is defined to be the asymptotic exponential growth rate of the sum $\sum_{i_1,\ldots,i_n=1}^N \|A_{i_n}\cdots A_{i_1}\|^p$, where p…

For a polynomial $P_n$ of degree $n$, Bernstein's inequality states that $\|P_n'\| \le n \|P_n\|$ for all $L^p$ norms on the unit circle, $0<p\le\infty,$ with equality for $P_n(z)= c z^n.$ We study this inequality for random polynomials,…

Let $p(z)=a_0+a_1z+a_2z^2+a_3z^3+\cdots+a_nz^n$ be a polynomial of degree $n,$ where the coefficients $a_j,$ $j \in \{0,1,2,\cdots n\},$ may be complex. We impose some restriction on the coefficients of the real part of the given polynomial…

Given a polynomial $x \in {\mathbb R}^n \mapsto p(x)$ in $n=2$ variables, a symbolic-numerical algorithm is first described for detecting whether the connected component of the plane sublevel set ${\mathcal P} = \{x : p(x) \geq 0\}$…

We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…

If a bounded domain can be covered by the polydisk through a rational proper holomorphic map, then the Bergman projection is $L^p$-bounded for $p$ in a certain range depending on the ramified rational covering. This result can be applied to…

In this paper, we present an isomorphism between the ring of general polynomials over a division ring of degree $p$ over its center $F$ and the group ring of the free monoid with $p^2$ variables. Using this isomorphism, we define the…

A polynomial is expansive if all of its roots lie outside the unit circle. We define some special determinants involving the coefficients of a real polynomial and formulate necessary and sufficient conditions for expansivity using these…

We address some questions concerning indecomposable polynomials and their behaviour under specialization. For instance we give a bound on a prime $p$ for the reduction modulo $p$ of an indecomposable polynomial $P(x)\in \Zz[x]$ to remain…

We study the dynamics of polynomial-like mappings in several variables. A special case of our results is the following theorem. Let f be a proper holomorphic map from an open set U onto a Stein manifold V, $U\subset\subset V$. Assume f is…

We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let $p$ and $q$ be two independent random polynomials of degree $n$, whose roots are chosen independently from the probability…

We analyze the numerical range of high-dimensional random matrices, obtaining limit results and corresponding quantitative estimates in the non-limit case. For a large class of random matrices their numerical range is shown to converge to a…

We study the random \'etale algebra generated by a random polynomial with i.i.d. coefficients distributed according to Haar measure normalized on $\mathbb{Z}_p$. We determine the probability that this random algebra is an unramified field,…

It has been shown that the edge structure of the characteristic imset polytope is closely connected to the question of causal discovery. The diameter of a polytope is an indicator of how connected the polytope is and moreover gives us a…

The $p$-spectral radius of a graph $G=(V,E)$ with adjacency matrix $A$ is defined as $\lambda^{(p)}(G)=\max \{x^TAx : \|x\|_p=1 \}$. This parameter shows remarkable connections with graph invariants, and has been used to generalize some…

A probabilistic characterization of the dominance partial order on the set of partitions is presented. This extends work in "Symmetric polynomials and symmetric mean inequalities". Electron. J. Combin., 20(3): Paper 34, 2013. Let $n$ be a…

Given a permutation $\sigma$, the Robinson-Schensted correspondence determines a certain partition called the shape of $\sigma$. Famously, the shape measures the longest unions of increasing and decreasing subsequences, thus giving global…

We prove a sharp inequality for the modulus of the logarithmic derivative of a polynomial in the lemniscate components containing no critical points.