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Consider the representations of an algebraic group G. In general, polynomial invariant functions may fail to separate orbits. The invariant subring may not be finitely generated, or the number and complexity of the generators may grow…
This manuscript is intended as an accompaniment to Guth's "A restriction estimate using polynomial partitioning". We begin by summarizing the core ideas of the proof, elaborating the history and development of the techniques therein. From…
Let ${\mathbb F}_q$ be a finite field of characteristic two and ${\mathbb F}_q(X_1,...,X_n)$ a rational function field. We use matrix methods to obtain explicit transcendental bases of the invariant subfields of orthogonal groups and…
In this paper we present an equivalent statement to the Jacobian conjecture. For a polynomial map F on an affine space of dimension n, we define recursively n finite sequences of polynomials. We give an equivalent condition to the…
Criteria are given for determining whether an irreducible sextic equation with rational coefficients is algebraically solvable over the complex numbers.
Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on…
We sharpen to nearly optimal the known asymptotic and explicit bounds for the number of $\mathbb{F}_q$-rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic…
We consider systems of strict multivariate polynomial inequalities over the reals. All polynomial coefficients are parameters ranging over the reals, where for each coefficient we prescribe its sign. We are interested in the existence of…
Using the functional interpretation from proof theory, we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings. We extract effective bounds, some of which are new to the literature,…
We give conditions on a finite set of series of rational numbers to ensure that they are algebraically independent. Specialising our results to polynomials of lower degree, we also obtain new results on irrationality and $mathbb{Q}$-linear…
In this article we study polynomial logarithmic $q$-forms on a projective space and characterize those that define singular foliations of codimension $q$. Our main result is the algebraic proof of their infinitesimal stability when $q=2$…
The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness,…
We show the finiteness of perfect powers in orbits of polynomial dynamical systems over an algebraic number field. We also obtain similar results for perfect powers represented by ratios of consecutive elements in orbits. Assuming the…
We discuss the formal aspects of the factorial polynomials and of the associated series. We develop the theory using the formalism of quasi-monomials and prove the usefulness of the method for the solutions of nontrivial difference…
Recently, a $q$-polynomial approach to the construction and analysis of cyclic codes over $\gf(q)$ was given by Ding and Ling. The objective of this paper is to give another $q$-polynomial approach to all cyclic codes over $\gf(q)$.
We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the…
This paper develops the geometry of locally bounded rational functions on non-singular real algebraic varieties. First various basic geometric and algebraic results regarding these functions are established in any dimension, culminating…
This paper presents a framework for abstracting uncertain or non-polynomial components of dynamical systems using polynomial constraints. This enables the application of polynomial-based analysis tools, such as sum-of-squares programming,…
We prove a lower bound on the canonical height associated to polynomials over number fields evaluated at points with infinite forward orbit. The lower bound depends only on the degree of the polynomial, the degree of the number field, and…
One proves a far-reaching upper bound for the degree of a generically finite rational map between projective varieties over a base field of arbitrary characteristic. The bound is expressed as a product of certain degrees that appear…