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We consider (near-)critical percolation on the square lattice. Let M_n be the size of the largest open cluster contained in the box [-n,n]^2, and let pi(n) be the probability that there is an open path from O to the boundary of the box. It…

We consider the problem of computing critical points of the restriction of a polynomial map to an algebraic variety. This is of first importance since the global minimum of such a map is reached at a critical point. Thus, these points…

Polynomials whose coefficients, roots, and critical points lie in the ring of rational integers are called nice polynomials. In this paper, we present a general method for investigating such polynomials. We extend our results from the ring…

We present a simple proof to a fact recently established in [5]: let $\xi$ be a symmetric random variable that has variance $1$, let $\Gamma=(\xi_{ij})$ be an $N \times n$ random matrix whose entries are independent copies of $\xi$, and set…

For a probability measure $\mu$ on $[0,1]$ without discrete component, the best possible order of approximation by a finite point set in terms of the star-discrepancy is $\frac{1}{2N}$ as has been proven relatively recently. However, if…

We say that a random integer variable $X$ is monotone if the modulus of the characteristic function of $X$ is decreasing on $[0,\pi]$. This is the case for many commonly encountered variables, e.g., Bernoulli, Poisson and geometric random…

A critical value of a function is the value of this function at one of its critical points. Each critical point of a differentiable multivariate function is described by the equations which consist in equating to zero all of its partial…

We prove that there is a universal measure on the unit circle such that any probability measure on the unit disk is the limit distribution of some subsequence of the corresponding orthogonal polynomials. This follows from an extension of a…

We consider the problem of bounding away from 0 the minimum value m taken by a polynomial P of Z[X_1,...,X_k] over the standard simplex, assuming that m>0. Recent algorithmic developments in real algebraic geometry enable us to obtain a…

We consider the problem of sensitivity of threshold risk, defined as the probability of a function of a random variable falling below a specified threshold level $\delta >0.$ We demonstrate that for polynomial and rational functions of that…

This note will study complex polynomial maps of degree $n\ge 2$ with only one critical point.

Consider a random sequence of $N$ integers, each chosen uniformly and independently from the set $\{1,\dots,x\}$. Motivated by applications to factorisation algorithms such as Dixon's algorithm, the quadratic sieve, and the number field…

Given a probability measure $\mu$ with infinite support on the unit circle $\partial\mathbb{D}=\{z:|z|=1\}$, we consider a sequence of paraorthogonal polynomials $\h_n(z,\lambda)$ vanishing at $z=\lambda$ where $\lambda \in \T$ is fixed. We…

In this paper, we show that for a unicritical polynomial having a priori bounds, the unique conformal measure of minimal exponent has no atom at the critical point. For a conformal measure of higher exponent, we give a necessary and…

We have discussed earlier the correlation functions of the random variables $\det(\la-X)$ in which $X$ is a random matrix. In particular the moments of the distribution of these random variables are universal functions, when measured in the…

In this paper we study the density of polynomials in some $L^2(M)$ spaces. Two choices of the measure $M$ and polynomials are considered: 1) a $(N\times N)$ matrix non-negative Borel measure on $\mathbb{R}$ and vector-valued polynomials…

Consider $n$ $d$-dimensional vectors with iid entries from a lattice distribution $X$. We show that the probability that all distances between them are equal is asymptotically \[ C_n\cdot\frac{1}{d^{(m-1)/2}} \quad \text{for} \quad d \to…

We prove for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n has the order of square root of n. Moment or symmetry assumptions are not necessary. In removing…

The (weak) Nullstellensatz over finite fields says that if $P_1,\ldots,P_m$ are $n$-variate degree-$d$ polynomials with no common zero over a finite field $\mathbb{F}$ then there are polynomials $R_1,\ldots,R_m$ such that…

For a Reinhardt domain $\Omega$ with the smooth boundary in $\mathbb{C}^{m+1}$ and a positive smooth measure $\mu$ on the boundary of $\Omega$, we consider the ensemble $P_{N}$ of polynomials of degree $N$ with the Gaussian probability…