Related papers: Efficient congruencing in ellipsephic sets: the ge…

In this paper, we bound the number of solutions to a quadratic Vinogradov system of equations in which the variables are required to satisfy digital restrictions in a given base. Certain sets of permitted digits, namely those giving rise to…

Let $I_{s,k,r}(X)$ denote the number of integral solutions of the modified Vinogradov system of equations $$x_1^j+\ldots +x_s^j=y_1^j+\ldots +y_s^j\quad (\text{$1\le j\le k$, $j\ne r$}),$$ with $1\le x_i,y_i\le X$ $(1\le i\le s)$. By…

We show that the system of equations \begin{align*} \sum_{i=1}^s (x_i^j-y_i^j) = a_j \qquad (1 \le j \le k) \end{align*} has appreciably fewer solutions in the subcritical range $s < k(k+1)/2$ than its homogeneous counterpart, provided that…

In this paper, we obtain an asymptotic formula for the number of integral solutions to a system of diagonal equations. We obtain an asymptotic formula for the number of solutions with variables restricted to smooth numbers as well. We…

Let $\mathbb{F}_q$ be a finite field with $q=p^n$ elements. In this paper, we study the number of solutions of equations of the form $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$ with $x_i\in\mathbb{F}_{p^{t_i}}$, where $b\in\mathbb{F}_q$ and…

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the boundary of the geometric (Minkowski) sum of $k$ ellipsoids in $n$-dimensional Euclidean space. Previously this was done…

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the boundary of the geometric (Minkowski) sum of $k$ ellipsoids in $n$-dimensional Euclidean space. Previously this was done…

We prove bounds for the number of solutions to $$a_1 + \dots + a_k = a_1' + \dots + a_k'$$ over $N$-element sets of reals, which are sufficiently convex or near-convex. A near-convex set will be the image of a set with small additive…

We use new bounds of double exponential sums with ratios of integers from prescribed intervals to get an asymptotic formula for the number of solutions to congruences $$ \sum_{j=1}^n a_j x_jy_j^{-1} \equiv a_0 \pmod p, $$ with variables…

We obtain asymptotic upper bounds for the number of natural solutions of the following diagonal Diophantine equations in a hypercube with side - $N$ in the paper: $x_1 = x_2^k+...+x_s^k$, $x_1^k = x_2^k+...+x_s^k$, $x_1 = \sum_{j=2}^s…

We show that whenever $s>k(k+1)$, then for any complex sequence $(\mathfrak a_n)_{n\in \mathbb Z}$, one has $$\int_{[0,1)^k}\left| \sum_{|n|\le N}\mathfrak a_ne(\alpha_1n+\ldots +\alpha_kn^k) \right|^{2s}\,{\rm d}{\mathbf \alpha}\ll…

We show that there are fewer than (e^2+3) 2^(k choose 2) n^k/4 non-degenerate positive solutions to a fewnomial system consisting of n polynomials in n variables having a total of n+k+1 distinct monomials. This is significantly smaller than…

We establish improved mean value estimates associated with the number of integer solutions of certain systems of diagonal equations, in some instances attaining the sharpest conjectured conclusions. This is the first occasion on which…

In this short paper, we give an upper bound for the number of different basic feasible solutions generated by the simplex method for linear programming problems having optimal solutions. The bound is polynomial of the number of constraints,…

We deal with a weakly coupled system of ODEs of the type $$ x_j'' + n_j^2 \,x_j + h_j(x_1,\ldots,x_d) = p_j(t), \qquad j=1,\ldots,d, $$ with $h_j$ locally Lipschitz continuous and bounded, $p_j$ continuous and $2\pi$-periodic, $n_j \in…

An asymptotic formula for the number of prime solutions of a general diagonal system of Diophantine equations is established, contingent on the existence of an appropriate mean value bound and on local solvability. In conjunction with the…

Covering numbers are a powerful tool used in the development of approximation algorithms, randomized dimension reduction methods, smoothed complexity analysis, and others. In this paper we prove upper bounds on the covering number of…

When $k\ge 4$ and $0\le d\le (k-2)/4$, we consider the system of Diophantine equations \[ x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\le j\le k,\, j\ne k-d). \] We show that in this cousin of a Vinogradov system, there is a paucity of…

We consider a weakly coupled singularly perturbed variational elliptic system in a bounded smooth domain with Dirichlet boundary conditions. We show that, in the competitive regime, the number of fully nontrivial solutions with nonnegative…

We study systems of $n \geq 1$ discrete differential equations of order $k\geq1$ in one catalytic variable and provide a constructive and elementary proof of algebraicity of their solutions. This yields effective bounds and a systematic…