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We consider properties of polynomials with coefficients in division rings. A theorem on the decomposition of a polynomial with coefficients in an arbitrary division ring is obtained. It is shown that if a non-central element is not a root…

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many…

We extend the Ax-Katz theorem for a single polynomial from finite fields to the rings Z_m with m composite. This extension not only yields the analogous result, but gives significantly higher divisibility bounds. We conjecture what computer…

If $\vf_1, ... \vf_m\colon\Z\to\Z^\ell$ are polynomials with zero constant terms and $E\subset\Z^\ell$ has positive upper Banach density, then we show that the set $E\cap (E-\vf_1(p-1))\cap\...\cap (E-\vf_m(p-1))$ is nonempty for some prime…

In this article, we propose a few sufficient conditions on polynomials having integer coefficients all of whose zeros lie outside a closed disc centered at the origin in the complex plane and deduce the irreducibility over the ring of…

We use spectral theory and algebraic geometry to establish a higher-degree analogue of a Szemer\'edi--Trotter-type theorem over finite fields, with an application to polynomial expansion.

The Schinzel hypothesis is a famous conjectural statement about primes in value sets of polynomials, which generalizes the Dirichlet theorem about primes in an arithmetic progression. We consider the situation that the ring of integers is…

We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory we give a proof of a result of Helmke, Jordan and Lieb on the number of linear unimodular matrix polynomials over a finite…

In this paper we obtain some statements concerning ideals of polynomials and apply these results in a number of different situations. Among other results, we present new characterizations of $\mathcal{L}_{\infty}$-spaces, Coincidence…

We prove a variant of the multidimensional polynomial Szemer\'edi theorem of Bergelson and Leibman where one replaces polynomial sequences with other sparse sequences defined by functions that belong to some Hardy field and satisfy certain…

The ring of integer-valued polynomials on an arbitrary integral domain is well-studied. In this paper we initiate and provide motivation for the study of integer-valued polynomials on commutative rings and modules. Several examples are…

We describe the ideals, especially the prime ideals, of semirings of polynomials over layered domains, and in particular over supertropical domains. Since there are so many of them, special attention is paid to the ideals arising from…

We prove that a sumset of a TE subset of (\N) (these sets can be viewed as "aperiodic" sets) with a set of positive upper density intersects a set of values of any polynomial with integer coefficients., i.e. for any (A \subset \N ) a TE…

Let $z_1, \dots, z_m$ be $m$ distinct complex numbers, normalized to $|z_k| = 1$, and consider the polynomial $$ p_{m}(z) = \prod_{k=1}^{m}{(z-z_k)}.$$ We define a sequence of polynomials in a greedy fashion, $$ p_{N+1}(z) = p_{N}(z)…

Let $ K $ be a number field, $ S $ a finite set of places of $ K $, and $ \mathcal{O}_S $ be the ring of $ S $-integers. Moreover, let $$ G_n^{(0)} Z^d + \cdots + G_n^{(d-1)} Z + G_n^{(d)} $$ be a polynomial in $ Z $ having simple linear…

The paper studies some properties of the ring of integer-valued quasi-polynomials. On this ring, theory of generalized Euclidean division and generalized GCD are presented. Applications to finite simple continued fraction expansion and…

We demonstrate that the phenomenon of popular differences (aka the phenomenon of large intersections) holds for natural families of polynomial patterns in rings of integers of number fields. If $K$ is a number field with ring of integers…

Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal…

We classify the polynomials with integral coefficients that, when evaluated on a group element of finite order $n$, define a unit in the integral group ring for infinitely many positive integers $n$. We show that this happens if and only if…

Let D be a domain with quotient field K and A a D-algebra. We call a polynomial with coefficients in K that maps every element of A to an element of A "integer-valued on A". For commutative A we also consider integer-valued polynomials in…