Related papers: Analytic number theory for 0-cycles

Using the technique developed in approximation theory, we construct examples of exponential families of infinitely divisible laws which can be viewed as deformations of the normal, gamma, and Poisson exponential families. Replacing the…

We generalize the polynomial Szemer\'{e}di theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new…

Let $q$ be a prime power, let $\mathbb F_q$ be the finite field with $q$ elements and let $\theta$ be a generator of the cyclic group $\mathbb F_q^*$. For each $a\in \mathbb F_q^*$, let $\log_{\theta} a$ be the unique integer $i\in \{1,…

We study sums of Dirichlet characters over polynomials in $\mathbb{F}_q[t]$ with a prescribed number of irreducible factors. Our main results are explicit formulae for these sums in terms of zeros of Dirichlet L-functions. We also exhibit…

We generalize the classical lifting and recombination scheme for rational and absolute factorization of bivariate polynomials to the case of a critical fiber. We explore different strategies for recombinations of the analytic factors,…

Classes of simple polynomial and simple trigonometric splines given by Fourier series are considered. It is shown that the class of simple trigonometric splines includes the class of simple polynomial splines. For some parameter values, the…

We introduce, characterise and provide a combinatorial interpretation for the so-called $q$-Jacobi-Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order $q$-differential…

Let $F$ be any field containing the finite field of order $q$. A $q$-polynomial $L$ over $F$ is an element of the polynomial ring $F[x]$ with the property that all powers of $x$ that appear in $L$ with nonzero coefficient have exponent a…

We present a few factorizations of polynomials over finite fields. These factorizations are related to traces, compositions of polynomials and binomial coefficients. As a corollary we obtain a description of all irreducible polynomials…

Some polynomials $P$ with rational coefficients give rise to well defined maps between cyclic groups, $\Z_q\longrightarrow\Z_r$, $x+q\Z\longmapsto P(x)+r\Z$. More generally, there are polynomials in several variables with tuples of rational…

Counterparts of several classical results of number theory are proven for the ring of polynomials with coefficients in a number field. A theorem of Milnor that determines the Witt ring of a function field is applied to prove an analogue of…

We start with a (q,t)-generalization of a binomial coefficient. It can be viewed as a polynomial in t that depends upon an integer q, with combinatorial interpretations when q is a positive integer, and algebraic interpretations when q is…

We compare various groups of 0-cycles on quasi-projective varieties over a field. As applications, we show that for certain singular projective varieties, the Levine-Weibel Chow group of 0-cycles coincides with the corresponding…

For nonzero coprime integers $a$ and $b$, a positive integer $\ell$ is said to be \emph{good with respect to $a$ and $b$} if there exists a positive integer $k$ such that $\ell$ divides $a^{k} + b^{k}$. The concept of good integers has been…

For a bridgeless graph $G$, its flow polynomial is defined to be the function $F(G,q)$ which counts the number of nonwhere-zero $\Gamma$-flows on an orientation of $G$ whenever $q$ is a positive integer and $\Gamma$ is an additive Abelian…

In this article we compute the $q$th power values of the quadratic polynomials $f$ with negative squarefree discriminant such that $q$ is coprime to the class number of the splitting field of $f$ over $\mathbb{Q}$. The theory of unique…

The following result, a consequence of Dumas criterion for irreducibility of polynomials over integers, is generally proved using the notion of Newton diagram: Let $f(x)$ be a polynomial with integer coefficients and $k$ be a positive…

We investigate generalised polynomials (i.e. polynomial-like expressions involving the use of the floor function) which take the value $0$ on all integers except for a set of density $0$. Our main result is that the set of integers where a…

Recently, a $q$-polynomial approach to the construction and analysis of cyclic codes over $\gf(q)$ was given by Ding and Ling. The objective of this paper is to give another $q$-polynomial approach to all cyclic codes over $\gf(q)$.

Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting…