Related papers: Restricted Variable Chevalley-Warning Theorems

In this paper we introduce a new approach and obtain new results for the problem of studying polynomial images of affine subspaces of finite fields. We improve and generalise several previous known results, and also extend the range of such…

We obtain upper bounds, independent of the ambient dimension, for the number of realizable zero-nonzero patterns and (over ordered fields) sign conditions of a finite family of polynomials $\mathcal P$ restricted to an algebraic subset $V$…

We prove several bounds on the number of incidences between two sets of multivariate polynomials of bounded degree over finite fields. From these results, we deduce bounds on incidences between points and multivariate polynomials, extending…

We begin by explaining how arguments used by R. Wilson to give an elementary proof of the $\mathbb F_p$ case for the Ax-Katz Theorem can also be used to prove the following generalization of the Chevalley-Warning and Ax-Katz Theorems for…

The ring of invariant polynomials ${\mathbb C}[V]^G$ over a given finite dimensional representation space $V$ of a complex reductive group $G$ is known, by a famous theorem of Hilbert, to be finitely generated. The general proof being…

A regularity lemma for polynomials provides a decomposition in terms of a bounded number of approximately independent polynomials. Such regularity lemmas play an important role in numerous results, yet suffer from the familiar shortcoming…

We derive optimal asymptotic and non-asymptotic lower bounds on the Widom factors for weighted Chebyshev and orthogonal polynomials on compact subsets of the real line. In the Chebyshev case we extend the optimal non-asymptotic lower bound…

It is shown that a trivial version of polarization is sufficient to produce separating systems of polynomial invariants: if two points in the direct sum of the $G$--modules $W$ and $m$ copies of $V$ can be separated by polynomial…

We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space Q_k of quasi-invariants of a given multiplicity is not, in general, an algebra but a module over the…

We prove a version of the Chevalley Restriction Theorem for the action of a real reductive group G on a topological space X which locally embeds into a holomorphic representation. Assuming that there exists an appropriate quotient X//G for…

A sum of a large-dimensional random matrix polynomial and a fixed low-rank matrix polynomial is considered. The main assumption is that the resolvent of the random polynomial converges to some deterministic limit. A formula for the limit of…

We survey results on Chebyshev polynomials centered around the work of H. Widom. In particular, we discuss asymptotics of the polynomials and their norms and general upper and lower bounds for the norms. Several open problems are also…

We characterize the limiting distributions of random variables of the form $P_n\left( (X_i)_{i \ge 1} \right)$, where: (i) $(P_n)_{n \ge 1}$ is a sequence of multivariate polynomials, each potentially involving countably many variables;…

Consider a sequence of polynomials of bounded degree evaluated in independent Gaussian, Gamma or Beta random variables. We show that, if this sequence converges in law to a nonconstant distribution, then (i) the limit distribution is…

In the present note we prove a conjecture of Demailly for finite sets of sufficiently many very general points in projective spaces. This gives a lower bound on Waldschmidt constants of such sets. Waldschmidt constants are asymptotic…

We study lower bound estimates for the number of solutions of systems of equations over finite fields. Heath-Brown improved the lower bounds given by the classical \emph{Chevalley-Warning Theorems} by excluding systems of equations whose…

We extend the method of Ghasemi and Marshall [SIAM. J. Opt. 22(2) (2012), pp 460-473], to obtain a lower bound $f_{{\rm gp},M}$ for a multivariate polynomial $f(x) \in \mathbb{R}[x]$ of degree $ \le 2d$ in $n$ variables $x = (x_1,...,x_n)$…

The quasi likelihood analysis is generalized to the partial quasi likelihood analysis. Limit theorems for the quasi likelihood estimators, especially the quasi Bayesian estimator, are derived in the situation where existence of a slow…

We study totally bounded subsets in weighted variable exponent amalgam and Sobolev spaces. Moreover, this paper includes several detailed generalized results of some compactness criterions in these spaces.

Let V be a complex vector space with basis {x_1,x_2,...,x_n} and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x_1, x_2,..., x_n with complex…