Integral-valued rational functions on valued fields

Let K be a field and a non-trivial valuation ring of K withm as its maximal ideal. Denote by and the rings of polynomials f∈K[X] and rational functions f∈K(X) resp. such that . We prove that for one variable X we have if and only if the completion of (K, ) is locally compact or algebraically closed. In the second case—i.e. if K is dense in the algebraic closure of (K, )—we even get for any number of variables X=(X₁,...,X_n). This work contains parts of the second author's thesis [Ri] written under the supervision of the first author.     

Hilbert's Tenth Problem for fields of rational functions over finite fields

Summary We prove that there is no algorithm to solve arbitrary polynomial equations over a field of rational functions in one letter with constants in a finite field of characteristic other than 2 and hence, Hilbert's Tenth Problem for any such field is undecidable.Oblatum 1-XI-1989Supported in part by NSF Grant DMS 8605198.     

Splitting fields of conics and sums of squares of rational functions

Given a geometrically unirational variety over an infinite base field, we show that every finite separable extension of the base field that splits the variety is the residue field of a closed point. As an application, we obtain a characterization of function fields of smooth conics in which every sum of squares is a sum of two squares.     

Reduced unitary Whitehead groups of skew fields of noncommutative rational functions

We calculate the reduced unitary Whitehead groups of skew fields of quotients of noncommutative polynomial rings. We prove a stability theorem in the case where the noncommutative polynomial ring is related to an inner automorphism of the original skew field.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 142–148, 1979.     

Distributions on rational function fields

Mathematische Annalen -     

Arithmetic properties of values of analytic functions connected by algebraic equations over fields of rational functions

In this paper theorems are proved about the arithmetic character of the values at algebraic points of a collection of G-functions which constitute a solution of a system of linear differential equations with coefficients from C(z) connected by algebraic equations over C(z). In addition, the theorems on E-functions proved by A. B. Shidlovskii in 1962 are supplemented.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 83–94, July, 1973.In conclusion, I express thanks to A. B. Shidlovskii for attention to and help with this work.     

Models of Peano Arithmetic and a question of Sikorski on ordered fields

Using models of Peano Arithmetic, we solve a problem of Sikorski by showing that the existence of an ordered field of cardinalityλ with the Bolzano-Weierstrass property forκ-sequences is equivalent to the existence of aκ-tree with exactlyλ branches and with noκ-Aronszajn subtrees. Supported in part by NSF Grant MCS-8301603.     

Derivations and linear functions along rational functions

The main purpose of this paper is to give characterization theorems on derivations as well as on linear functions. Among others the following problem will be investigated: Let ${n \in \mathbb{Z}, f, g\colon\mathbb{R} \to\mathbb{R}}$ be additive functions, ${\left(\begin{array}{cc} a&b\\ c&d \end{array} \right) \in \mathbf{GL}_{2}(\mathbb{Q})}$ be arbitrarily fixed, and let us assume that the mapping $$ \phi(x)=g\left(\frac{ax^{n}+b}{cx^{n}+d}\right)-\frac{x^{n-1}f(x)}{(cx^{n}+d)^{2}} \quad \left(x\in\mathbb{R}, cx^{n}+d\neq 0\right)$$ satisfies some regularity on its domain (e.g. (locally) boundedness, continuity, measurability). Is it true that in this case the above functions can be represented as a sum of a derivation and a linear function? Analogous statements ensuring linearity will also be presented.     

Pair correlation of roots of rational functions with rational generating functions and quadratic denominators

For any rational functions with complex coefficients A(z),B(z), and C(z), where A(z), C(z) are not identically zero, we consider the sequence of rational functions H _m (z) with generating function ∑H _m (z)t ^m =1/(A(z)t ²+B(z)t+C(z)). We provide an explicit formula for the limiting pair correlation function of the roots of $\prod_{m=0}^{n}H_{m}(z)$ , as n→∞, counting multiplicities, on certain closed subarcs J of a curve $\mathcal{C}$ where the roots lie. We give an example where the limiting pair correlation function does not exist if J contains the endpoints of $\mathcal{C}$ .     

An answer to a question on the convolution of functions

Let G be a locally compact group. Recently, G?a¸b and Strobin [2] asked when f*g exists for all \({f, g \in L^p(G)}\) , and also: is the set \({\{(f,g)\in L^p(G) \times L^p(G): f\ast g \in L^p(G)\}}\) σ-c-lower porous (in particular, meager) for \({p\in(1,2]}\) ? In this paper, we answer these questions. In particular, we prove that if 1 <  p <  ∞, 1 ≤  q < ∞, and G is a non-unimodular locally compact group, then the set \({\{(f, g) \in L^p(G) \times L^q(G): f * g}\) is not λ -a.e. finite on G} is a residual set in L ^p (G) ×  L ^q (G).     

Twisted rational functions and series

We study rings of twisted rational functions and twisted Laurent series over simple artinian rings. We determine the centers of such rings and investigate the structure of subalgebras of these rings. We extend to infinite dimensional division rings and to simple artinian rings results proven in a previous paper for finite dimensional division algebras. We investigate “Galois subrings” of rings of fractions.     

Hecke-symmetry and rational period functions

In this paper, we continue our work in the direction of a characterization of rational period functions on the Hecke groups. We examine the role that Hecke-symmetry of poles plays in this setting, and pay particular attention to non-symmetric irreducible systems of poles for a rational period function. This gives us a new expression for a class of rational period functions of any positive even integer weight on the Hecke groups. We illustrate these properties with examples of specific rational period functions. We also correct the wording of a theorem from an earlier paper.     

Quadrature and orthogonal rational functions

Classical interpolatory or Gaussian quadrature formulas are exact on sets of polynomials. The Szegő quadrature formulas are the analogs for quadrature on the complex unit circle. Here the formulas are exact on sets of Laurent polynomials. In this paper we consider generalizations of these ideas, where the (Laurent) polynomials are replaced by rational functions that have prescribed poles. These quadrature formulas are closely related to certain multipoint rational approximants of Cauchy or Riesz–Herglotz transforms of a (positive or general complex) measure. We consider the construction and properties of these approximants and the corresponding quadrature formulas as well as the convergence and rate of convergence.