Related papers: Multi-variable Polynomial Solutions to Pell's Equa…

We consider sums of the form \[\sum_{j=0}^{n-1}F_1(a_1n+b_1j+c_1)F_2(a_2n+b_2j+c_2)... F_k(a_kn+b_kj+c_k),\] in which each $\{F_i(n)\}$ is a sequence that satisfies a linear recurrence of degree $D(i)<\infty$, with constant coefficients. We…

For each positive integer $n$, let $g_\Delta(n)$ be the smallest positive integer $g$ such that every complete quadratic polynomial in $n$ variables which can be represented by a sum of odd squares is represented by a sum of at most $g$ odd…

Suppose $k$ is a positive integer and $\mathcal{X}$ is a $k$-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most $k$ sets. Suppose there is a function…

Let $n>1$ and let $R$ be a commutative ring with identity $1\ne 0$ and $R[x_1,\ldots,x_n]^n$ the set of all $n$-tuples of polynomials of the form $(f_1,\ldots,f_n),$ where $f_1,\ldots,f_n\in R[x_1,\ldots,x_n]$. We call these $n$-tuples…

Let $\mathbf{f} = (f_1, \ldots, f_R)$ be a system of polynomials with integer coefficients in which the degrees need not all be the same. We provide sufficient conditions for which the system of equations $f_j (x_1, \ldots, x_n) = 0 \ (1…

Given a differential operator T=\sum_{i=1}^k Q_i(z)d^i/dz^i where each Q_i(z) is a polynomial define r=max_i deg(Q_i(z)-i). Assuming that r is nonnegative we consider the following multiparameter spectral problem: for each positive integer…

Let $F_1,\ldots,F_R$ be homogeneous polynomials with integer coefficients in $n$ variables with differing degrees. Write $\boldsymbol{F}=(F_1,\ldots,F_R)$ with $D$ being the maximal degree. Suppose that $\boldsymbol{F}$ is a nonsingular…

Let $g>1$ be an integer and $f(X)\in{\mathbb Z}[X]$ a polynomial of positive degree with no multiple roots, and put $u(n)=f(g^n)$. In this note, we study the sequence of quadratic fields ${\mathbb Q}(\sqrt{u(n)}\,)$ as $n$ varies over the…

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…

The solution to the general univariate polynomial equation has been sought for centuries. It is well known there is no general solution in radicals for degrees five and above. The hyper-Catalan numbers $C[m_2,m_3,m_4,\ldots]$ count the ways…

Permutation polynomials over finite fields have extensive applications in various areas. Particularly, permutation polynomials with simple forms are of great interest. In recent papers, several classes of permutation polynomials of the form…

We study the problem of computing the isolated regular solutions of a system \((f_1,\ldots,f_n)\) of \(n\) polynomial equations in \(n\) variables \((X_1, \dots, X_n)\) over a field of characteristic zero \(k\). We focus on systems with a…

Given $k\in N$, a nonnegative function $f\in C^r[a,b]$, $r\ge 0$, an arbitrary finite collection of points $\big\{\alpha_i\big\}_{i\in J} \subset [a,b]$, and a corresponding collection of nonnegative integers $\big\{m_i\big\}_{i\in J}$ with…

Let $a, b, c,$ and $n$ be integers, with $a$ nonzero and $n$ at least two. Necessary and sufficient conditions on these parameters are derived which guarantee that all solutions of the congruence \[ ax^2+bx+c \equiv 0\ \textrm{mod}\ n \]…

In this paper we prove that decomposable forms, or homogeneous polynomials $F(x_1, \cdots, x_n)$ with integer coefficients which split completely into linear factors over $\mathbb{C}$, take on infinitely many square-free values subject to…

We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible ${\mathbb{F}}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in…

Given a positive noncommutative polynomial $f$, equivalently a sum of Hermitian squares (SOHS), there exists a positive semidefinite Gram matrix that encrypts all the structural essence of $f$. There are no available methods for extending a…

We study a generalization of the classical correspondence between homogeneous quadratic polynomials, quadratic forms, and symmetric/alternating bilinear forms to forms in $n$ variables. The main tool is combinatorial polarization, and the…

Let $p$ be a prime number, $m$ be an even positive integer, and $\mathbb{F}_q$ be a finite field with $q = p^m$ elements. In this paper, we compute the number of solutions with all coordinates in $\mathbb{F}_q^*$ for diagonal equations of…

We classify the polynomials with integral coefficients that, when evaluated on a group element of finite order $n$, define a unit in the integral group ring for infinitely many positive integers $n$. We show that this happens if and only if…