Related papers: Small solutions to linear forms in primes

Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a…

An integral binary quartic form is said to be locally soluble (resp. soluble) if the corresponding genus one curve has a rational point over $\mathbb{Q}_v$ for every place $v$ of $\mathbb{Q}$ (resp. over $\mathbb{Q}$). We consider the…

We discuss linear autonomous evolution equations on function spaces which have the property that a positive initial value leads to a solution which initially changes sign, but then becomes - and stays - positive again for sufficiently large…

By adopting a new approach to the analysis of the density of p-adic solutions arising in applications of the circle method, we show that under modest conditions the existence of non-trivial p-adic solutions suffices to establish positivity…

We give a specific method to solve with quadratic complexity the linear systems arising in known algorithms to deal with the sign determination problem. In particular, this enable us to improve the complexity bound for sign determination in…

The main goal of this note is to establish the limits of L. Zhao's techniques for counting solutions to quadratic forms in prime variables. Zhao considered forms with rank at least 9, and showed that these equations have solutions in primes…

In this sequel to Resolution except for minimal singularities I, we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singularities except for normal crossings. The main new feature is…

In this paper we generalize the result of Fouvry and Iwaniec dealing with prime values of the quadratic form $x^2 + y^2$ with one input restricted to a thin subset of the integers. We prove the same result with an arbitrary primitive…

Let $Q$ be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer $n$ by $Q$. This problem is connected with deriving an upper bound on…

We show the existence of infinitely many positive solutions, defined on the real line, for the nonlinear scalar ODE \[ \ddot u + (a^+(t) - \mu a^-(t)) u^3 = 0, \] where $a$ is a periodic, sign-changing function, and the parameter $\mu>0$ is…

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime square and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in…

A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical…

We study the solubility of cubic equations over the integers. Assuming a necessary congruence condition, the existence of such solutions is established when the $h$-invariant of $C$ is at least $14$, improving on work of Davenport-Lewis and…

We obtain a lower bound for a number of primes in tuples. As applications, we obtain a lower bound for the Romanoff type representation functions.

Over an algebraically closed field, we describe the affine varieties of solutions to the linear equations $a(xb)=c$ and $a(bx)=c$ over the split-octonions. We also determine the dimensions of the solution sets of arbitrary linear monomial…

This paper describes infinite sets of polynomial equations in infinitely many variables with the property that the existence of a solution or even an approximate solution for every finite subset of the equations implies the existence of a…

We prove asymptotic formulae for small weighted solutions of quadratic congruences of the form $\lambda_1x_1^2+\cdots +\lambda_nx_n^2\equiv \lambda_{n+1}\bmod{p^m}$, where $p$ is a fixed odd prime, $\lambda_1,...,\lambda_{n+1}$ are integer…

We prove that suitable asymptotic formulae in short intervals hold for the problems of representing an integer as a sum of a prime and a square, or a prime square. Such results are obtained both assuming the Riemann Hypothesis and in the…

We give improved lower bounds for the number of solutions of some $S$-unit equations over the integers, by counting the solutions of some associated linear equations as the coefficients in those equations vary over sparse sets. This method…

Let k => 1, m => 1 be small fixed integers, gcd(k, m) = 1. This note develops some techniques for proving the existence of infinitely many primes solutions x = p, and y = q of the linear Diophantine equation y = mx + k.