Related papers: Birch's theorem with shifts

Consider the following Kirchhoff type problem $$ \left\{\aligned -\bigg(a+b\int_{\mathbb{B}_R}|\nabla u|^2dx\bigg)\Delta u&= \lambda u^{q-1} + \mu u^{p-1}, &\quad \text{in}\mathbb{B}_R, \\ u&>0,&\quad\text{in}\mathbb{B}_R,\\…

In this study, we investigate the form of solutions, stability character and asymptotic behavior of the following rational difference equation x_{n+1}=({\gamma}/(x_{n}(x_{n-1}+{\alpha})+\b{eta})), n=0,1,..., where the inital values x_{-1}…

Let $\mathcal{P}$ be the set of primes and $\mathbb{N}$ the set of positive integers. Let also $r_1,...,r_t$ be positive real numbers and $R_2(r_1,...,r_t)$ the set of odd integers which can be represented as $$ p+2^{\lfloor…

For a set $A$ of non-negative integers, let $R_A(n)$ denote the number of solutions to the equation $n=a+a'$ with $a$, $a'\in A$. Denote by $\chi_A(n)$ the characteristic function of $A$. Let $b_n>0$ be a sequence satisfying $\limsup_{n\to…

We show that for each n-tuple of positive rational integers (a_1,..,a_n) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a_1x_1+...+a_nx_n=1 with the x_i all S-units are not contained in…

Let $F_1,\ldots,F_R$ be homogeneous polynomials of degree $d\ge 2$ with integer coefficients in $n$ variables, and let $\mathbf{F}=(F_1,\ldots,F_R)$. Suppose that $F_1,\ldots,F_R$ is a non-singular system and $n\ge 4^{d+2}d^2R^5$. We prove…

Let $d,k$ be natural numbers and let $\mathcal{L}_1, \dots, \mathcal{L}_k \in \mathrm{GL}_d(\mathbb{Q})$ be linear transformations such that there are no non-trivial subspaces $U, V \subseteq \mathbb{Q}^d$ of the same dimension satisfying…

Let $\lambda_i, \mu_j$ be non-zero real numbers not all of the same sign and let $a_i, b_k$ be non-zero integers not all of the same sign. We investigate a mixed Diophantine system of the shape \begin{equation*} \begin{cases} \left|…

For cyclic totally real number fields $K$ with odd prime degree $n$, odd class number, $2$ inert, and the property that every totally positive unit is a square, the density of rational primes $p$ that satisfy the spin relation…

We use the circle method to count the number of integer solutions to systems of bihomogeneous equations of bidegree $(1,1)$ and $(2,1)$ of bounded height in lopsided boxes. Previously, adjusting Birch's techniques to the bihomogeneous…

We study the number of representations of an integer n=F(x_1,...,x_s) by a homogeneous form in sufficiently many variables. This is a classical problem in number theory to which the circle method has been succesfully applied to give an…

Given an infinite sequence of positive integers $\cA$, we prove that for every nonnegative integer $k$ the number of solutions of the equation $n=a_1+...+a_k$, $a_1,\,..., a_k\in \cA$, is not constant for $n$ large enough. This result is a…

Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$. Let $R_F(Z)$ denote the number of integers of absolute value at most $Z$ which are represented by $F$. We prove that there is a…

Given an integral indefinite binary Hermitian form f over an imaginary quadratic number field, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational…

We study solutions $(x_n)_{n \in \mathbb{N}}$ of nonhomogeneous nonlinear second order difference equations of the type $\ell_n = x_n ( \sigma_{n,1} x_{n+1} + \sigma_{n,0} x_n + \sigma_{n,-1} x_{n-1} ) + \kappa_n x_n$, with given initial…

Fix a prime $p >2$ and a finite field $\mathbb{F}_{q}$ with $q$ elements, where $q$ is a power of $p$. Let $m$ be a monic polynomial in the polynomial ring $\mathbb{F}_{q}[T]$ such that $deg(m)$ is large. Fix an integer $r\geq 2$, and let…

We consider a system of $R$ cubic forms in $n$ variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided $n\geq 25R$, we prove an asymptotic formula for the number of integer points in…

Let $\mathbb{F}_q$ stand for the finite field of odd characteristic $p$ with $q$ elements ($q=p^{n},n\in \mathbb{N} $) and $\mathbb{F}_q^*$ denote the set of all the nonzero elements of $\mathbb{F}_{q}$. Let $m$ and $t$ be positive…

A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form $x$, $x+d$, $x+d^2$. We obtain a multidimensional version of this result, which can be regarded as a first step towards…

We establish an aysmptotic formula for the number of points with coordinates in $\mb{F}_q[t]$ on a complete intersection of degree $d$ defined over $\mb{F}_q[t]$, with explicit error term, provided that the characteristic of $\mb{F}_q$ is…