For every finitary endofunctor H of Set a rational algebraic theory (or a rational finitary monad) R is defined by means of solving all finitary flat systems of recursive equations over H. This generalizes the result of Elgot and his coauthors, describing a free iterative theory of a polynomial endofunctor H as the theory R of all rational infinite trees. We present a coalgebraic proof that R i...

Let V be a K3 surface defined over a number field k. The BatyrevManin conjecture for V states that for every nonempty open subset U of V , there exists a finite set ZU of accumulating rational curves such that the density of rational points on U − ZU is strictly less than the density of rational points on ZU . Thus, the set of rational points of V conjecturally admits a stratification correspon...

Let p be a fixed prime number. Throughout this paper, the symbols Z, Zp, Qp, and Cp denote the ring of rational integers, the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N ∪ {0}. The p-adic absolute value in Cp is normalized in such a way that |p|p 1/p see 1–17 . Fo...

We investigate when a rational endomorphism of the Riemann sphere C extends to a mapping of the upper half-space H3 which is rational with respect to some measurable conformal structure. Such an extension has the property that it and all its iterates have uniformly bounded distortion. Such maps are called uniformly quasiregular. We show that, in the space of rational mappings of degree d, such ...

One of the “big problems” of number theory is to understand the set of rational points on a variety, or equivalently, the rational solutions to a system of polynomial equations. Despite thousands of years of research, we are very far from having a general method for solving all such problems. There is even some evidence that deciding the existence of a rational solution is an undecidable proble...

We describe examples of rational maps which are not topologically conjugate to a polynomial and whose Julia sets are connected but not locally connected. Introduction and motivations The dynamics of a rational map f acting on Ĉ is concentrated on its Julia set which is (by definition) the minimal compact set invariant by f and f−1 containing at least three points. The question of local connecti...

Nathaniel Dean asks the following: Is it possible to find four nonconcyclic points on the parabola y = x2 such that each of the six distances between pairs of points is rational? We demonstrate that there is a correspondence between all rational points satisfying this condition and orbits under a particular group action of rational points on a fiber product of (three copies of) an elliptic surf...

This paper explores the nature of solution sets systems equations in virtually abelian groups. We view this question from two angles. From a formal language perspective, we prove that set solutions to system forms an EDT0L language, with respect natural normal form. Looking at growth, show growth series is rational. Furthermore, considering as tuples group elements, it has rational relative any...

In this paper, we provide a new scheme for approximating the weakly efficient solution set class of vector optimization problems with rational objectives over feasible defined by finitely many polynomial inequalities. More precisely, present procedure to obtain sequence explicit approximations problem in question. Each approximation is intersection sublevel single and set. To end, make use achi...