Related papers: A dynamical argument for a Ramsey property

We resolve the Ramsey problem for $\{x,y,z:x+y=p(z)\}$ for all polynomials $p$ over $\mathbb{Z}$. In particular, we characterise all polynomials that are $2$-Ramsey, that is, those $p(z)$ such that any $2$-colouring of $\mathbb{N}$ contains…

We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_1h(x_1) + \cdots + a_sh(x_s)=b$, where the $a_i$ and $b$ are fixed coefficients, and $h$ is an arbitrary integer…

S\'ark\"ozy proved that dense sets of integers contain two elements differing by a $k$th power. The bounds in quantitative versions of this theorem are rather weak compared to what is expected. We prove a version of S\'ark\"ozy's theorem…

Let $\alpha$ and $\beta$ be irrational real numbers and $0<\F<1/30$. We prove a precise estimate for the number of positive integers $q\leq Q$ that satisfy $\|q\alpha\|\cdot\|q\beta\|<\F$. If we choose $\F$ as a function of $Q$ we get…

This paper provides a survey of particular values of Ramanujan's theta function $\varphi(q)=\sum_{n=-\infty}^{\infty}q^{n^2}$, when $q=e^{-\pi\sqrt{n}}$, where $n$ is a positive rational number. First, descriptions of the tools used to…

Fix positive integers $p$ and $q$ with $2 \leq q \leq {p \choose 2}$. An edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ different colors. The function $f(n, p, q)$ is the…

A polynomial with rational coefficients is said to be pure with respect to a rational prime $p$ if its Newton polygon has one slope. In this article, we prove that the number of irreducible factors of the $n$-th iterate of a pure polynomial…

A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: for any prime $p$, reduce its coefficients mod $p$ and consider its action on the field $\mathbb{F}_p$. We say a subset of…

For $q \in (0, 1)$, the deformed exponential function $f(x) = \sum_{n \geq 1} x^n q^{n(n-1)/2}/n!$ is known to have infinitely many simple and negative zeros $\{x_k(q)\}_{k \geq 1}$. In this paper, we analyze the series expansions of…

Recent years have witnessed the introduction and development of extremely fast rational function algorithms. Many ideas in this realm arose from polynomial-based linear-algebraic algorithms. However, polynomial approximation is occasionally…

For the quantum integer [n]_q = 1+q+q^2+... + q^{n-1} there is a natural polynomial multiplication such that [mn]_q = [m]_q \otimes_q [n]_q. This multiplication is given by the functional equation f_{mn}(q) = f_m(q) f_n(q^m), defined on a…

We resolve a conjecture of Cooper-Fenner-Purewal that a certain sequence of combinatorial matrices which can be used to bound small product-Ramsey numbers is positive semidefinite. Because the connection to Ramsey Theory involves solving…

We introduce a bilateral extension of the continuous $q$-ultraspherical polynomials which we call bilateral $q$-ultraspherical functions. These functions are given as specific bilateral basic hypergeometric ${}_2\psi_2$ series, they are…

Theorem. An irreducible cubic polynomial with rational coefficients has a root in a one step radical extension of Q if and only if the discriminate is a square of a rational number. Theorem. An irreducible polynomial x^4+px^2+qx+s with…

$q$-Analogues of the coefficients of $x^a$ in the expansion of $\prod_{j=1}^N (1+x+...+x^j)^{L_j}$ are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the ``$q$-supernomial coefficients'' are…

We discuss $Q(n)$, the number of ways a given integer $n$ may be written as a sum of distinct primes, and study its asymptotic form $Q_{as}(n)$ valid in the limit $n\to\infty$. We obtain $Q_{as}(n)$ by Laplace inverting the fermionic…

We conjecture a formula for the rational $q,t$-Catalan polynomial $\mathcal{C}_{r/s}$ that is symmetric in $q$ and $t$ by definition. The conjecture posits that $\mathcal{C}_{r/s}$ can be written in terms of symmetric monomial strings…

Given a prime power $q$ and positive integers $m,t,e$ with $e > mt/2$, we determine the number of all monic irreducible polynomials $f(x)$ of degree $m$ with coefficients in $\mathbb{F}_q$ such that $f(x^t)$ contains an irreducible factor…

In this paper we characterize the subspace of $\mathcal{L}_{q,1,v}$ of function which are the q-Bessel Fourier transform of positive functions in $\mathcal{L}_{q,1,v}$. As application we give a q-version of the Bochner's theorem.

There is a natural pluripotential-theoretic extremal function V_{K,Q} associated to a closed subset K of C^m and a real-valued, continuous function Q on K. We define random polynomials H_n whose coefficients with respect to a related…