Related papers: A Rationality Problem of Some Cremona Transformati…

We show that the coefficients of rational 2-functions are contained in an abelian number field. More precisely, we show that the poles of such functions are poles of order one and given by roots of unity and rational residue.

Let $F$ be a field and let $F(X_1,\dots,X_n)$ be the field of rational functions in $n$ variables $X_1,\dots,X_n$ over $F$. Let $T=X_1+\cdots+X_n\in F(X_1,\dots,X_n)$ and let $m$ be a positive integer such that $\text{char}\,F\nmid m$. Is…

Generating functions for a fixed genus map and hypermap enumeration become rational after a simple explicit change of variables. Their numerators are polynomials with integer coefficients that obey a differential recursion, and denominators…

Given a rational function of degree at least two defined over a number field k, we study the cardinality of the set of rational iterated preimages. We prove bounds for the cardinality of this set as the rational function varies in certain…

We show that even dimensional Fermat cubic hypersurfaces are rational over any field of characteristic different from three by producing explicit rational parametrizations given by polynomials of low degree. As a byproduct of our…

In this paper we present an algorithm to compute all unirational fields of transcendence degree one containing a given finite set of multivariate rational functions. In particular, we provide an algorithm to decompose a multivariate…

Let $X$ be a finite, 2-dimensional cell complex. The curvature invariants $\rho_\pm(X)$ and $\sigma_\pm(X)$ were defined in [13], and a programme of conjectures was outlined. Here, we prove the foundational result that the quantities…

F. Campana had asked whether a certain threefold is rational. In arXiv:1310.3569v1 [mathAG], this variety was shown to be birational to a specific conic bundle and then to be unirational. We prove that this conic bundle is rational.

In this note we prove a result comparing rationality of algebraic cycles over the function field of a $SL_1(A)$-torsor for a central simple algebra $A$ and over the base field.

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 will prove that the zeta function for Ruelle-expanding maps is rational.

Feynman integral computations in theoretical high energy particle physics frequently involve square roots in the kinematic variables. Physicists often want to solve Feynman integrals in terms of multiple polylogarithms. One way to obtain a…

We formulate explicitly the necessary and sufficient conditions for the local invertibility of a field transformation involving derivative terms. Our approach is to apply the method of characteristics of differential equations, by treating…

The arithmetic function of two variables is defined. Some properties of the function are given along with the formula that is an analog of the so-called Mobius' inversion formula. A heuristic statement is suggested.

We study the complexity of deterministic and probabilistic inversions of partial computable functions on the reals.

Suppose that $F$ is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve with positive self-intersection. We prove that if there exists a non-constant meromorphic function on $F$, then the…

The importance of transformations and normal forms in logic programming, and generally in computer science, is well documented. This paper investigates transformations and normal forms in the context of Defeasible Logic, a simple but…

We study rationality constructions for smooth complete intersections of two quadrics over nonclosed fields. Over the real numbers, we establish a criterion for rationality in dimension four.

We study examples where conformal invariance implies rational critical indices, triviality of the underlying quantum field theory and emergence of hypergeometric functions as solutions of the field equations.

Motivated by the Shapiro Shapiro conjecture, we consider the following: given a field $k$, under what conditions must a rational function with only $k$-rational ramification points be equivalent (after post-composition with a fractional…