Related papers: Sums of squares in function fields over Henselian …

We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to…

We are interested in the algebraic properties of groups of local biholomorphisms and their consequences. A natural question is whether the complexity of solvable groups is bounded by the dimension of the ambient space. In this spirit we…

In this sequence of work we investigate polynomial equations of additive functions. We consider the solutions of equation \[ \sum_{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x)^{q_{i}}= 0 \qquad \left(x\in \mathbb{F}\right), \] where $n$ is a positive…

We improve a bound due to the second author on number of rational points on smooth surfaces in $\mathbb{P}^3$ over finite fields and look at families of surfaces that achieve or nearly achieve this bound, for which we compute their exact…

We give effective bounds for the class number of any algebraic function field of genus $g$ defined over a finite field. These bounds depend on the possibly partial information on the number of places on each degree $\leq g$. Such bounds are…

To represent positive integers by regular patterns on a plane or in three-dimensional space may be traced back to the Pythagoreans. The aim of the present article is to explore the possibility of extending the representation framework for…

We consider Dirichlet $L$-functions $L(s, \chi)$ where $\chi$ is a non-principal quadratic character to the modulus $q$. We make explicit a result due to Pintz and Stephens by showing that $|L(1, \chi)|\leq \frac{1}{2}\log q$ for all $q\geq…

Inspired by the theory of Hodge correlators due to Goncharov and by the plectic principle of Nekov\'a\v{r} and Scholl, we construct higher plectic Green functions and give a higher order generalization of Hecke's formula for abelian…

The ring of dual numbers over a ring $R$ is $R[\alpha] = R[x]/(x^2)$, where $\alpha$ denotes $x+(x^2)$. For any finite commutative ring $R$, we characterize null polynomials and permutation polynomials on $R[\alpha]$ in terms of the…

For any real $k\geq 2$ and large prime $q$, we prove a lower bound on the $2k$-th moment of the Dirichlet character sum \begin{equation*} \frac{1}{\phi(q)} \sum_{\substack{\chi \text{ mod }q\\ \chi\neq \chi_0}} \Big| \sum_{n\leq x}…

In this survey, we explore how superorthogonality amongst functions in a sequence $f_1,f_2,f_3,\ldots$ results in direct or converse inequalities for an associated square function. We distinguish between three main types of…

Functionals (i.e. functions of functions) are widely used in quantum field theory and solid-state physics. In this paper, functionals are given a rigorous mathematical framework and their main properties are described. The choice of the…

We provide new upper bounds for sums of certain arithmetic functions in many variables at polynomial arguments and, exploiting recent progress on the mean-value of the Erd\H os-Hooley $\Delta$-function, we derive lower bounds for the…

The sum of square roots is as follows: Given $x_1,\dots,x_n \in \mathbb{Z}$ and $a_1,\dots,a_n \in \mathbb{N}$ decide whether $ E=\sum_{i=1}^n x_i \sqrt{a_i} \geq 0$. It is a prominent open problem (Problem 33 of the Open Problems Project),…

Let $f$ be a polynomial of degree $d>1$ in $n$ variables over $\mathbb{Z}$. Let $f_d$ be the homogeneous part of degree $d$ of $f$ and $s$ be the dimension of the critical locus of $f_d$. In this paper, we prove Igusa's conjecture for…

In 1988, Andrews, Dyson and Hickerson initiated the study of q-hypergeometric series whose coefficients are dictated by the arithmetic in real quadratic fields. In this paper, we provide a dozen q-hypergeometric double sums which are…

This paper examines level sets of functions of the form $f(x)=\sum_{n=0}^\infty \frac{r_n}{2^n}\phi(2^n x)$, where phi(x) is the distance from x to the nearest integer, and r_n equals 1 or -1 for each n. Such functions are referred to as…

The size function for a number field is an analogue of the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. It was conjectured to attain its maximum at the trivial class of Arakelov divisors. This conjecture was…

Let X be a homogeneous tree of degree q+1 (for q between 2 and infinity) and let f be a complex function on X times X for which f(x,y) only depend on the distance between x and y in X. Our main result gives a necessary and sufficient…

Let $f_n(k)$ be the number of subrings of index $k$ in $\mathbb{Z}^n$. We show that results of Brakenhoff imply a lower bound for the asymptotic growth of subrings in $\mathbb{Z}^n$, improving upon lower bounds given by Kaplan, Marcinek,…