Related papers: D-finiteness, rationality, and height

We describe the structure of all codimension-two lattice configurations $A$ which admit a stable rational $A$-hypergeometric function, that is a rational function $F$ all whose partial derivatives are non zero, and which is a solution of…

We prove decidability of univariate real algebra extended with predicates for rational and integer powers, i.e., $(x^n \in \mathbb{Q})$ and $(x^n \in \mathbb{Z})$. Our decision procedure combines computation over real algebraic cells with…

We prove a formula for the Taylor series coefficients of a zero of the sum of a complex-exponent polynomial and a base function which is a general holomorphic function with a simple zero. Such a Taylor series is more general than a Puiseux…

Ufnarovski remarked in 1990 that it is unknown whether any finitely presented associative algebra of linear growth is automaton, that is, whether the set of normal words in the algebra form a regular language. If the algebra is graded, then…

Let k be a number field. It is well known that the set of sequences composed by Taylor coefficients of rational functions over k is closed under component-wise operations, and so it can be equipped with a ring structure. A conjecture due to…

Let $K_0(\mathcal{V}_{K})$ be the Grothendieck ring of varieties over a field $K$ of characteristic zero, and let $\mathbb{L} = [\mathbb{A}^1_{K}]$ denote the Lefschetz class. We prove that if a $K$-variety has $\mathbb{L}$-rational…

Let X be a smooth hypersurface of degree d in P^n over an algebraically closed field of characteristic p. We show that X must be separably rationally connected and must contain a free line if either p is at least d or if p is at least d-1…

In 1947, M. S. Macphail constructed a series in $\ell_{1}$ that converges unconditionally but does not converge absolutely. According to the literature, this result helped Dvoretzky and Rogers to finally answer a long standing problem of…

Let $X$ be an algebraic variety over a finite field $\bF_q$, homogeneous under a linear algebraic group. We show that the number of rational points of $X$ over $\bF_{q^n}$ is a periodic polynomial function of $q^n$ with integer…

A (noncommutative) P\'olya series over a field $K$ is a formal power series whose nonzero coefficients are contained in a finitely generated subgroup of $K^\times$. We show that rational P\'olya series are unambiguous rational series,…

To obtain the Dirichlet series for complex powers of the Riemann zeta function, we define and study the basic properties of a sequence of polynomials that, used as coefficients of the respective terms of the Dirichlet series of the Riemann…

Let $L$ be a nilpotent algebra of class two over a compact discrete valuation ring $A$ of characteristic zero or of sufficiently large positive characteristic. Let $q$ be the residue cardinality of $A$. The ideal zeta function of $L$ is a…

In this paper we study the set of rational solutions of equations defined by power sums symmetric polynomials with coefficients in a finite field. We do this by means of applying a methodology which relies on the study of the geometry of…

We determine the rationality properties of unipotent characters of finite reductive groups arising as fixed points of disconnected reductive groups under a Frobenius map. In the proof we use realisations of characters in $\ell$-adic…

We show that a finite type duality group of dimension $d>2$ is the fundamental group of a $(d+3)$-manifold with rationally acyclic universal cover. We use this to find closed manifolds with rationally acyclic universal cover and some…

For a faithful linear representation $V$ of a finite group $G$ in coprime characteristic, we show that if the field Noether number $\beta_{\mathrm{field}}$ is the minimum $d$ such that the invariant polynomials of degree $\leq d$ generate…

Rational twisted power series over a (commutative) field are studied. We give several characterizations of such series, which are similar to the classical results concerning rational power series over a commutative field. In particular, we…

We develop a sequential-topological study of rational points of schemes of finite type over local rings typical in higher dimensional number theory and algebraic geometry. These rings are certain types of multidimensional complete fields…

Our overall goal is to unify and extend some results in the literature related to the approximation of generating functions of finite and infinite sequences over a field by rational functions. In our approach, numerators play a significant…

For an analytic function $f(z)=\sum_{k=0}^\infty a_kz^k$ on a neighbourhood of a closed disc $D\subset {\bf C}$, we give assumptions, in terms of the Taylor coefficients $a_k$ of $f$, under which the number of intersection points of the…