Related papers: On diagonal equations over finite fields

We develop a theory of sesquilinear forms over finite fields, investigating their representations via polynomials and coefficient matrices, along with classification results for these forms. Through their connection to quadratic forms, we…

Given a non-singular diagonal cubic hypersurface $X\subset\mathbb{P}^{n-1}$ over $\mathbb{F}_q(t)$ with $\mathrm{char} (\mathbb{F}_q)\neq 3$, we show that the number of rational points of height at most $|P|$ is $O(|P|^{3+\varepsilon})$ for…

Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…

We study the differential uniformity of the Wan-Lidl polynomials over finite fields. A general upper bound, independent of the order of the field, is established. Additional bounds are established in settings where one of the parameters is…

Let $\mathbb{F}_q$ denote the finite field of $q$ elements with characteristic $p$. Let $\mathbb{Z}_q$ denote the unramified extension of the $p$-adic integers $\mathbb{Z}_p$ with residue field $\mathbb{F}_q$. In this paper, we investigate…

We design and analyze an algorithm for computing solutions with coefficients in a finite field $\mathbb{F}_q$ of underdetermined systems defined over $\mathbb{F}_q$. The algorithm is based on reductions to zero-dimensional searches. The…

On a closed Riemannian manifold $(M^n ,g)$ with a proper isoparametric function $f$ we consider the equation $\Delta^2 u -\alpha \Delta u +\beta u = u^q$, where $\alpha$ and $\beta$ are positive constants satisfying that $\alpha^2 \geq 4…

In this paper we study about the existence of solutions of certain kind of non-linear differential and differential-difference equations. We give partial answer to a problem which was asked by chen et al. in [13].

We give sufficient conditions to determine the existence of nontrivial solutions to the Fermat equation $x^3+y^3=kz^3$ over $\mathbb{Q}(\sqrt{d})$ by constructing a relationship with the points on the elliptic curve $y^2=x^3-432d^3k^2$ over…

Je retracerai l'histoire des conjectures de Weil sur le nombre de solutions d'\'equations polynomiales dans un corps fini et quelques unes des approches qui ont \'et\'e propos\'ees pour les r\'esoudre. The Weil conjectures: origins,…

Using modularity, level lowering, and explicit computations with Hilbert modular forms, Galois representations and ray class groups, we show that for $3 \le d \le 23$ squarefree, $d \ne 5$, $17$, the Fermat equation $x^n+y^n=z^n$ has no…

This paper investigates the upper bound of the number of integer (natural) solutions of inhomogeneous algebraic Diophantine diagonal equations with integer coefficients without a free member via the circle method of Hardy and Littlewood.…

In this note we obtain the solutions of four $q$-functional equations and express the solutions in $q$-operator forms. These equations give sufficient conditions for $q$-operator methods.

Let $S$ be a smooth cubic surface over a finite field $\mathbb F_q$. It is known that $\#S(\mathbb F_q) = 1 + aq + q^2$ for some $a \in \{-2,-1,0,1,2,3,4,5,7\}$. Serre has asked which values of a can arise for a given $q$. Building on…

We provide polynomial upper bounds on the size of a shortest solution for quadratic equations in a free group. A similar bound is given for parametric solutions in the description of solutions sets of quadratic equations in a free group.

We explore variants of Erd\H os' unit distance problem concerning dot products between successive pairs of points chosen from a large finite subset of either $\mathbb F_q^d$ or $\mathbb Z_q^d,$ where $q$ is a power of an odd prime.…

We study the Abel differential equation x0 = A(t)x3 + B(t)x2 +C(t)x. Specifically, we find bounds on the number of its rational solutions when A(t), B(t) and C(t) are polynomials with real or complex coefficients; and on the number of…

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints,…

We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can…

We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$, defined over a field $K$, where the $x_i$ are themselves homogeneous polynomials of some degree $d$ in $r+1$ variables. Equivalently, we are looking at rational…