Related papers: Translates of Polynomials

A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollob\'as-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted…

For any finite field $\mathbb{F}$ and any positive integer $n$ we count the number of monic polynomials of degree $n$ over $\mathbb{F}$ with nonzero constant coefficient and a self-reciprocal factor of any specified degree. An application…

In this note we present a description of wave front evolving from an algebraic hypersurface by means of a pull-back of the discriminantal loci of a tame polynomial via a polynomial mapping. As an application we give examples of wave fronts…

We consider sparse polynomials in $N$ variables over a finite field, and ask whether they vanish on a set $S^N$, where $S$ is a set of nonzero elements of the field. We see that if for a polynomial $f$, there is $\mathbf{c}\in S^N$ with $f…

In this paper, we study a second order variational problem for locally convex hypersurfaces, which is the affine invariant analogue of the classical Plateau problem for minimal surfaces. We prove existence, regularity and uniqueness results…

These are notes of a talk based on the work arXiv:1212.3630 joint with A. Aizenbud. Let V be a finite-dimensional vector space over a local field F of characteristic 0. Let f be a function on V of the form $f(x)= \psi (P(x))$, where P is a…

For a permutation f of an n-dimensional vector space V over a finite field of order q we let k-affinity(f) denote the number of k-flats X of V such that f(X) is also a k-flat. By k-spectrum(n,q) we mean the set of integers k-affinity(f)…

If K/k is a function field in one variable of positive characteristic, we describe a general algorithm to factor one-variable polynomials with coefficients in K. The algorithm is flexible enough to find factors subject to additional…

Let Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset…

We give an upper bound for the number of points of a hypersurface over a finite field that has no lines on, in terms of the dimension, the degree, and the number of the elements of the finite field.

We show that the intersection of the irreducible components of a hypersurface defined by a polynomial with square-free support has F-rational singularities in characteristic $p>0$. As a consequence, we obtain that hypersurfaces defined by…

In this note we look at the freeness for complex affine hypersurfaces. If $X \subset \mathbb{C}^n$ is such a hypersurface, and $D$ denotes the associated projective hypersurface, obtained by taking the closure of $X$ in $\mathbb{P}^n$, then…

The focus of the paper is on the maximal dimension of affine subspaces of nilpotent $n \times n $ matrices with fixed rank. In particular we obtain two results in the "border" cases rank equal to $n-1$ and rank equal to $1$.

We describe the action of a plane interface between two semi-infinite media in terms of a transfer matrix. We find a remarkably simple factorization of this matrix, which enables us to express the Fresnel coefficients as a hyperbolic…

We consider the function $x^{-1}$ that inverses a finite field element $x \in \mathbb{F}_{p^n}$ ($p$ is prime, $0^{-1} = 0$) and affine $\mathbb{F}_{p}$-subspaces of $\mathbb{F}_{p^n}$ such that their images are affine subspaces as well. It…

We study the dynamics of polynomial mappings f:C^k to C^k of degree at least 2 that extend continuously to projective space P^k. Our approach is to study the dynamics near the hyperplane at infinity and then making a descent to K --- the…

In the half-space model of hyperbolic space, that is, $\r^3_{+}=\{(x,y,z)\in\r^3;z>0\}$ with the hyperbolic metric, a translation surface is a surface that writes as $z=f(x)+g(y)$ or $y=f(x)+g(z)$, where $f$ and $g$ are smooth functions. We…

We use tools of additive combinatorics for the study of subvarieties defined by {\it high rank} families of polynomials in high dimensional $\mathbb{F} _q$-vector spaces. In the first, analytic part of the paper we prove a number properties…

To a generic hypersurface in the affine torus $(\mathbb{C}^*)^n$ we associate a hypersurface arrangement in the projective space $\mathbb{P}^n$ consisting of the $n+1$ coordinate hyperplanes and a generic hypersurface, and compute the…

In this paper, we prove a monotonicity formula and some Bernstein type results for translating solitons of hypersurfaces in $\re^{n+1}$, giving some conditions under which a trantranslating soliton is a hyperplane. We also show a gap…