Related papers: Prime solutions to polynomial equations in many va…

The bivariate difference filed $(\mathbb{F}(\alpha, \beta), \sigma)$ provides an algebraic framework for a sequence satisfying a recurrence of order two and it could transform the summation involving a sequence satisfying a recurrence of…

This paper resolves the question of pointwise convergence for ergodic averages of a single function along the set of polynomial values of primes of the form $x^2 + ny^2$. Following the influential paper of Bourgain…

We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the…

Let $f,g_1,\dots,g_m$ be polynomials of degree at most $d$ with real coefficients in a vector of variables $x=(x_1,\dots,x_n)$. Assume that $f$ is non-negative on a basic semi-algebraic set $S$ defined by polynomial inequalities $g_j(x)\ge…

This note investigates the prime values of the polynomial $f(t)=qt^2+a$ for any fixed pair of relatively prime integers $ a\geq 1$ and $ q\geq 1$ of opposite parity. For a large number $x\geq1$, an asymptotic result of the form $\sum_{n\leq…

Let us fix a prime $p$ and a homogeneous system of $m$ linear equations $a_{j,1}x_1+\dots+a_{j,k}x_k=0$ for $j=1,\dots,m$ with coefficients $a_{j,i}\in\mathbb{F}_p$. Suppose that $k\geq 3m$, that $a_{j,1}+\dots+a_{j,k}=0$ for $j=1,\dots,m$…

Let f in Z[x,y] be a reducible homoegeneous polynomial of degree 3. We show that f(x,y) has an even number of prime factors as often as an odd number of prime factors.

Let $f(x,y)$ be a real polynomial of degree $d$ with isolated critical points, and let $i$ be the index of $grad f$ around a large circle containing the critical points. An elementary argument shows that $|i| \leq d-1$. In this paper we…

Let f be an irreducible polynomial of degree d>=3 with no fixed prime divisor. We derive an asymptotic formula for the number of primes p<x such that f(p) is (d-1)-free.

Conditions for the existence of at least three positive solutions to the nonlinear first-order problem with a nonlinear nonlocal boundary condition given by && y'(t) - p(t)y(t) = \sum_{i=1}^m f_i\big(t,y(t)\big), \quad t\in[0,1], && \lambda…

The Alexander-Hirschowitz theorem says that a general collection of $k$ double points in ${\bf P}^n$ imposes independent conditions on homogeneous polynomials of degree $d$ with a well known list of exceptions. We generalize this theorem to…

We investigate the zeros of polynomial solutions to the differential-difference equation \[ P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x), n=0,1,... \] where $A_{n}$ and $B_{n}$ are polynomials of degree at most 2 and 1…

The complexity of computing the solutions of a system of multivariate polynomial equations by means of Groebner bases computations is upper bounded by a function of the solving degree. In this paper, we discuss how to rigorously estimate…

A 1993 result of Alon and F\"uredi gives a sharp upper bound on the number of zeros of a multivariate polynomial over an integral domain in a finite grid, in terms of the degree of the polynomial. This result was recently generalized to…

Let $r$ be a prime power and $q=r^m$. For $0\le i\le m-1$, let $f_i\in \mathbb{F}_r[x]$ be $q$-linearized and $a_i\in \mathbb{F}_q$. Assume that $z\in \mathbb{\bar{F}}_r$ satisfies the equation $\sum_{i=0}^{m-1}a_if_i(z)^{r^i}=0$, where…

We consider the differential equations y''=\lambda_0(x)y'+s_0(x)y, where \lambda_0(x), s_0(x) are C^{\infty}-functions. We prove (i) if the differential equation, has a polynomial solution of degree n >0, then \delta_n=\lambda_n…

We establish a new class of examples of the multivariate Bateman-Horn conjecture by using tools from dynamics. These cases include the determinant polynomial on the space of $n\times n$ matrices, the Pfaffian on the space of skew-symmetric…

Let $f(t,y,y')=\sum_{i=0}^n a_i(t,y)y'^i=0$ be an irreducible first order ordinary differential equation with polynomial coefficients. Eremenko in 1998 proved that there exists a constant $C$ such that every rational solution of…

This paper presents a universal numerical scheme tailored for tackling linear integral, integro-differential, and both initial and boundary value problems of ordinary differential equations. The numerical scheme is readily adapted for…

We develop a version of the Hardy-Littlewood circle method to obtain asymptotic formulas for averages of general multivariate arithmetic functions evaluated at polynomial arguments in several variables. As an application, we count the…