Related papers: An algebraic geometry version of the Kakeya proble…

Given polynomial maps $f, g \colon \mathbb{R}^n \to \mathbb{R}^n,$ we consider the {\em polynomial complementary problem} of finding a vector $x \in \mathbb{R}^n$ such that \begin{equation*} f(x) \ \ge \ 0, \quad g(x) \ \ge \ 0, \quad…

Let $f(x_1,...,x_k)$ be a polynomial over a field $K$. This paper considers such questions as the enumeration of the number of nonzero coefficients of $f$ or of the number of coefficients equal to $\alpha\in K^*$. For instance, if $K=\ff_q$…

Let M be a quasiprojective algebraic manifold with K_M=0 and G a finite automorphism group of M acting trivially on the canonical class K_M; for example, a subgroup G of SL(n,C) acting on C^n in the obvious way. We aim to study the quotient…

We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms $f, g\in…

New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…

This work consists of two parts. In the first part we develop new techniques to compute Koszul cohomology groups for several classes of varieties. As applications we prove results on projective normality and syzygies for algebraic surfaces.…

In this paper, we prove the following non-linear generalization of the classical Sylvester-Gallai theorem. Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$, and $\mathcal{F}=\{F_1,\cdots,F_m\} \subset…

We study subsets of the $n$-dimensional vector space over the finite field $\mathbb{F}_q$, for odd $q$, which contain either a sphere for each radius or a sphere for each first coordinate of the center. We call such sets radii spherical…

Let $\cal{A}$ be the algebra of quaternions $\mathbb{H}$ or octonions $\mathbb{O}$. In this manuscript a new proof is given, based on ideas of Cauchy and D' Alembert, of the fact that an ordinary polynomial $f(t) \in {\cal{A}}\, [t]$ has a…

Let K be a field of characteristic zero. We prove that images of a linear K-derivation and a linear K-E-derivation of the ring K[x 1 ,x 2 ,x 3 ] of polynomial in three variables over K are Mathieu-Zhao subspaces, which affirms the LFED…

The cross covariogram g_{K,L} of two convex sets K, L in R^n is the function which associates to each x in R^n the volume of the intersection of K with L+x. The problem of determining the sets from their covariogram is relevant in…

Let $X,Y$ be two irreducible subvarieties of the projective space $\mathbb{P}^n$, and $d\geq 1$ an integer number. The main result of this paper is an algorithm to construct {\bf explicitly}, in terms of $d$ and the ideals defining $X$ and…

We introduce and study a new class of algebras, which we name \textit{quantum generalized Heisenberg algebras} and denote by $\mathcal{H}_q (f,g)$, related to generalized Heisenberg algebras, but allowing more parameters of freedom, so as…

Let X be a complex nonsingular affine algebraic variety, K a holomorphically convex subset of X, and Y a homogeneous variety for some complex linear algebraic group. We prove that a holomorphic map f:K-->Y can be uniformly approximated on K…

Let f\in \mathbb{Z}[x,y] be an irreducible homogeneous polynomial of degree 3. We show that f(x,y) has an even number of prime factors as often as an odd number of prime factors.

A conjecture of Odoni stated over Hilbertian fields $K$ of characteristic zero asserts that for every positive integer $d$, there exists a polynomial $f\in K[x]$ of degree $d$ such that for every positive integer $n$, each iterate $f^{\circ…

We present a uniform framework for constructing $3$-designs from $\mathrm{GL}_2(\mathbb F_q)$-invariant subspaces of $\mathbb F_q[X,Y]_k$, the space of homogeneous polynomials of degree $k$. Given such a subspace $W$, we associate a…

We prove the discrete analogue of Kakeya conjecture over $\mathbb{R}^n$. This result suggests that a (hypothetically) low dimensional Kakeya set cannot be constructed directly from discrete configurations. We also prove a generalization…

In this paper we present a decision procedure for computing pFq hypergeometric solutions for third order linear ODEs, that is, solutions for the classes of hypergeometric equations constructed from the 3F2, 2F2, 1F2 and 0F2 standard…

Let $\L (f) = K[x][y; f\frac{d}{dx} ]$ be an Ore extension of a polynomial algebra $K[x]$ over a field $K$ of characteristic zero where $f\in K[x]$. For a given polynomial $f$, the automorphism group of the algebra $\L (f) $ is explicitly…