Related papers: Density of integer solutions to diagonal quadratic…

Let $\mathbb{F}_q$ be a finite field of $q=p^k$ elements. For any $z\in \mathbb{F}_q$, let $A_n(z)$ and $B_n(z)$ denote the number of solutions of the equations $x_1^3+x_2^3+\cdots+x_n^3=z$ and $x_1^3+x_2^3+\cdots+x_n^3+zx_{n+1}^3=0$…

We study the existence of formal power series solutions to q-algebraic equations. When a solution exists, we give a sufficient condition on the equation for this solution to have a positive radius of convergence. We emphasize on the case…

We show that any subset of the squares of positive relative upper density contains non-trivial solutions to a translation-invariant linear equation in five or more variables, with explicit quantitative bounds. As a consequence, we establish…

We consider the integro-differential equation ${\rm I}^{\alpha}_{0+}f= x^m f$ on the half-line. We show that there exists a density solution, which is then unique and can be expressed in terms of the Beta distribution, if and only if $m>…

In this paper we obtain explicit estimates and existence results on the number of $\mathbb{F}_q$-rational solutions of certain systems defined by families of diagonal equations over finite fields. Our approach relies on the study of the…

We prove that quadratical quasigroups form a variety Q of right and left simple groupoids. New examples of quadratical quasigroups of orders 25 and 29 are given. The fine structure of quadratical quasigroups and inter-relationships between…

Let $f(x_1,\ldots,x_n)$ be a regular indefinite integral quadratic form with $n\ge 9$, and let $t$ be an integer. It is established that $f(x_1,\ldots,x_n)=t$ has solutions in prime variables if there are no local obstructions.

In this paper we focus on the unconstrained binary quadratic optimization model, maximize x^t Qx, x binary, and consider the problem of identifying optimal solutions that are robust with respect to perturbations in the Q matrix.. We are…

In this article, we construct explicit meromorphic solutions of first order linear $q$-difference equations in the complex domain and we describe the location of all their zeros and poles. The homogeneous case leans on the study of four…

An effective upper bound is established for the least non-trivial integer solution to the system of cubic forms \[ \begin{cases} F = c_{1}x_1^3 + c_{2}x_2^3 + \cdots + c_{n}x_n^3 = 0, \\ G = d_{1}x_1^3 + d_{2}x_2^3 + \cdots + d_{n}x_n^3 =…

In this paper, we consider the following diagonal quadratic forms \begin{equation*} a_1x_1^2 + a_2x_2^2 + \cdots + a_{\ell}x_{\ell}^2, \end{equation*} where $\ell\ge 5$ is an odd integer and $a_i\ge 1$ are integers. By using the extended…

This paper presents an integer decomposition method. The method first writes an integer as a polynomial with 2 as variable that its coefficients are zero or one. Then, suppose that an integer is decomposed into product of such two…

In this paper we study the problem of long gaps between values of binary quadratic forms. Let $D_{1}$, $D_{2},\ldots ,D_{r}$ be negative integers and $(s_{n})_{n=1}^{\infty}$ be the sequence of all the numbers representable by any binary…

In light of Kim's conjecture on regular polytopes of dimension four, which is a generalization of Waring's problem, we establish asymptotic formulas for representing any sufficiently large integer as a sum of numbers in the form of those…

In this paper, we find a basis for the space of modular forms of weight $2$ on $\Gamma_1(48)$. We use this basis to find formulas for the number of representations of a positive integer $n$ by certain quaternary quadratic forms of the form…

For $A\subseteq \{1, 2, \ldots\}$, we consider $R(A)=\{a/b: a, b\in A\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of nonzero values assumed by a cubic form. We study this problem…

Let $d_S$ denote the arithmetic density of a subset $S \subseteq \mathbb N$. We derive a power series in $q\in \mathbb C$, $|q|<1$, with co\"efficients related to integer partitions and integer compositions, that yields $1/d_S$ in the limit…

For polynomials of degree two which have no zeros, the method of accompanying variables is developed and zeros of associated vector polynomials are determined. Our flexible method uses a wide variety of possible vector-valued vector…

Given an isotropic quadratic form over a number field which assumes a value $t$, we investigate the distribution of points at which this value is assumed. Building on the previous work about the distribution of small-height zeros of…

This article is concerned with the Fourier coefficients of cusp forms (not necessarily eigenforms) of half-integer weight lying in the plus space. We give a soft proof that there are infinitely many fundamental discriminants $D$ such that…