Related papers: A note on the field isomorphism problem of X^3+sX+…

A perfect Euler cuboid is a rectangular parallelepiped with integer edges, with integer face diagonals, and with integer space diagonal as well. Finding such parallelepipeds or proving their non-existence is an old unsolved mathematical…

For the finite field $\mathbb{F}_{2^{3m}}$, permutation polynomials of the form $(x^{2^m}+x+\delta)^{s}+cx$ are studied. Necessary and sufficient conditions are given for the polynomials to be permutation polynomials. For this, the…

We obtain a polynomial type upper bound for the size of the integral solutions of Thue equations $F(X,Y) = b$ defined over a totally real number field $K$, assuming that $F(X,1)$ has a root $\alpha$ such that $K(\alpha)$ is a CM-field.…

A multi-cube method is developed for solving systems of elliptic and hyperbolic partial differential equations numerically on manifolds with arbitrary spatial topologies. It is shown that any three-dimensional manifold can be represented as…

Though it is well known that the roots of any affine polynomial over a finite field can be computed by a system of linear equations by using a normal base of the field, such solving approach appears to be difficult to apply when the field…

We are interested in solving the congruences $f^3+g^3+1\equiv 0\pmod{fg}$ and $f^4-4g^2+4\equiv 0\pmod{fg}$ in polynomials $f, g$ with rational coefficients. Moreover, we present results of computations of all integer points on certain one…

Let k=F_q be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non singular quartic plane curves defined over k. We find explicit rational normal models and we give closed…

We show that every monic polynomial of degree three with complex coefficients and no repeated roots is either a (vertical and horizontal) translation of $y=x^3$ or can be composed with a linear function to obtain a Ramanujan cubic. As a…

We consider multiple orthogonal polynomials associated with the exponential cubic weight e^{-x^3} over two contours in the complex plane. We study the basic properties of these polynomials, including the Rodrigues formula and…

We consider three types of rings of supersymmetric polynomials: polynomial ones $\Lambda_{m,n}$, partially polynomial $\Lambda_{m,n}^{+y}$ and Laurent supersymmetric rings $\Lambda_{m,n}^{\pm}$. For each type of rings we give their…

Let $\mathbb{F}_q$ be a finite field. Given two irreducible polynomials $f,g$ over $\mathbb{F}_q$, with $\mathrm{deg} f$ dividing $\mathrm{deg} g$, the finite field embedding problem asks to compute an explicit description of a field…

Let $n$ be a cubefree natural number and $p\geq 5$ be a prime number. Assume that $n$ is not expressible as a sum of the form $x^3+y^3$, where $x,y\in \mathbb{Q}$. In this note, we study the solutions (or lack thereof) to the equation…

We compute the Galois group of the splitting field $F$ of any irreducible and separable polynomial $f(x)=x^6+ax^3+b$ with $a,b\in K$, a field with characteristic different from two. The proofs require to distinguish between two cases:…

Let k be a quadratic field. We give an explicit formula for the Dirichlet series enumerating cubic fields whose quadratic resolvent field is isomorphic to k. Our work is a sequel to previous work of Cohen and Morra, where such formulas are…

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…

Let $K$ be a number field of degree $d\geq 3$ and fix $s$ multiplicatively independent algebraic integers $\gamma_1, \dots, \gamma_s \in K^*$ that fulfil some technical requirements, which can be vastly simplified to $\mathbb{Q}$-linearly…

Let $F_n$ denote the $n$-th Fibonacci number and $L_n$ the $n$-th Lucas number. We completely solve the family of cubic Thue equations $${(X-F_nY)(X-L_nY)X-Y^3=\pm1}$$ and show that there are no non-trivial solutions for $n\neq 1,3$.

We show the 3 by 3 magic square of squares problem equivalent to solving quartic polynomials with certain factorization constraints over an abelian extension of the rationals. We analyze a particular case in which said extension is assumed…

Generalised Pythagorean triples are integer tuples $(x,y,z)$ satisfying the equation $E_{a,b,c}: ax^2+by^2+cz^2=0$. A significant amount of research has been devoted towards understanding generalised Pythagorean triples and, in particular,…

In [U. Dempwolff: \textit{More Translation Planes and Semifields from Dembowski-Ostrom Polynomials}, Designs, Codes, Cryptogr. \textbf{68} (1-3) (2013), 81--103] three classes of rank two presemifields of order $q^{2n}$, with $q$ and $n$…