Related papers: A Polynomial Time Test to Detect Number with Many …

The Sylvester's denumerant \( d(t; \boldsymbol{a}) \) is a quantity that counts the number of nonnegative integer solutions to the equation \( \sum_{i=1}^{N} a_i x_i = t \), where \( \boldsymbol{a} = (a_1, \dots, a_N) \) is a sequence of…

For an algebraic number $\alpha$ we denote by $M(\alpha)$ the Mahler measure of $\alpha$. As $M(\alpha)$ is again an algebraic number (indeed, an algebraic integer), $M(\cdot)$ is a self-map on $\overline{\mathbb{Q}}$, and therefore defines…

1. There is no existing any quadratic interval $\eta_{n}:=(n^{2},(n+1)^{2}],$ which contains less than 2 prime numbers. The number of prime numbers within $\eta_{n}$ goes averagely linear with n to infinity. 2. The exact law of the number…

Under certain conditions, we give an estimate from above on the number of differential equations of order $r+1$ with prescribed regular singular points, prescribed exponents at singular points, and having a quasi-polynomial flag of…

It is well known that almost every dilation of a sequence of real numbers, that diverges to $\infty$, is dense modulo~1. This paper studies the exceptional set of points -- those for which the dilation is not dense. Specifically, we…

We consider the problem of efficiently solving a system of $n$ non-linear equations in ${\mathbb R}^d$. Addressing Smale's 17th problem stated in 1998, we consider a setting whereby the $n$ equations are random homogeneous polynomials of…

Let $m_t(\alpha)$ denote the $t$-metric Mahler measure of the algebraic number $\alpha$. Recent work of the first author established that the infimum in $m_t(\alpha)$ is attained by a single point $\bar\alpha = (\alpha_1,\ldots,\alpha_N)\in…

This is the latest in a series of articles aimed at exploring the relationship between the complexity classes of P and NP. In the previous papers, we have proved that the sat CNF problem is polynomially reduced to the problem of finding a…

Let $\Gamma\subset \overline{\mathbb Q}^{\times}$ be a finitely generated multiplicative group of algebraic numbers. Let $\delta, \beta\in\overline{\mathbb Q}^\times$ be algebraic numbers with $\beta$ irrational. In this paper, we prove…

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that…

Let xi be a real number which is neither rational nor quadratic over Q. Based on work of Davenport and Schmidt, Bugeaud and Laurent have shown that, for any real number theta, there exist a constant c>0 and infinitely many non-zero…

Consider real or complex polynomial Riccati differential equations $a(x) \dot y=b_0(x)+b_1(x)y+b_2(x)y^2$ with all the involved functions being polynomials of degree at most $\eta$. We prove that the maximum number of polynomial solutions…

This paper studies the saddle point problem of polynomials. We give an algorithm for computing saddle points. It is based on solving Lasserre's hierarchy of semidefinite relaxations. Under some genericity assumptions on defining…

We show that computing even very coarse approximations of critical points is intractable for simple classes of nonconvex functions. More concretely, we prove that if there exists a polynomial-time algorithm that takes as input a polynomial…

In this expository paper we describe four primality tests. The first test is very efficient, but is only capable of proving that a given number is either composite or 'very probably' prime. The second test is a deterministic polynomial time…

We study rather general multiple zeta-functions whose denominators are given by polynomials. The main aim is to prove explicit formulas for the values of those multiple zeta-functions at non-positive integer points. We first treat the case…

Evaluating a polynomial on a set of points is a fundamental task in computer algebra. In this work, we revisit a particular variant called trimmed multipoint evaluation: given an $n$-variate polynomial with bounded individual degree $d$ and…

In this paper we describe an algorithm that takes as input a description of a semi-algebraic set $S \subset \R^k$, defined by a Boolean formula with atoms of the form $P > 0, P < 0, P=0$ for $P \in {\mathcal P} \subset \R[X_1,...,X_k],$ and…

Let $\alpha$ be a real number such that $1< \alpha <2$ and let $x_0=x_0(\alpha)$ be a {\rm(}unique{\rm)} positive solution of the equation $$ x^{\alpha-1} -\frac{\pi}{e^2\sqrt{3}}x +1=0. $$ Then we prove that for each positive integer…

We consider the problem of uniform sampling of points on an algebraic variety. Specifically, we develop a randomized algorithm that, given a small set of multivariate polynomials over a sufficiently large finite field, produces a common…