Related papers: Places, cuts and orderings of function fields

Two fields are Witt equivalent if their Witt rings of symmetric bilinear forms are isomorphic. Witt equivalent fields can be understood to be fields having the same quadratic form theory. The behavior of finite fields, local fields, global…

We reduce the classification of finite extensions of function fields (of curves over finite fields) with the same class number to a finite computation; complete this computation in all cases except when both curves have base field…

It is well known that the composition of a D-finite function with an algebraic function is again D-finite. We give the first estimates for the orders and the degrees of annihilating operators for the compositions. We find that the analysis…

The problem of characterizing normed ordered spaces which admit a representation in the algebraic, order and norm sense as a subspace of $C(X)$, the space of all continuous functions on a compact Hausdorff space is a classical problem that…

In this paper, we study square functions for extension operators over finite-type, planar curves endowed with the Euclidean arclength measure. We prove new results for curves of the form $(T,\phi(T))$ where $\phi(T)$ is a polynomial of…

Any compact body in ${\mathbb R}^N$ with smooth boundary defines a two-valued function on the space of affine hyperplanes: the volumes of two parts into which these hyperplanes cut the body. This function is never algebraic if $N$ is even…

Given a smooth cubic hypersurface $X$ over a finite field of characteristic greater than 3 and two generic points on $X$, we use a function field analogue of the Hardy-Littlewood circle method to obtain an asymptotic formula for the number…

We review the following subjects: 1. Basic theory on algebraic curves and their moduli space, 2. Schottky uniformization theory of Riemann surfaces, and its extension called arithmetic uniformization theory, 3. Application to these theories…

We develop a theory of ordered *-vector spaces with an order unit. We prove fundamental results concerning positive linear functionals and states, and we show that the order (semi)norm on the space of self-adjoint elements admits multiple…

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…

One of the main themes in this thesis is the description of the signature of both the infinite place and the finite places in cubic function fields of any characteristic and quartic function fields of characteristic at least 5. For these…

This paper is devoted to the study of mappings in metric spaces. We investigate mappings satisfying inverse moduli inequalities. We show that under certain conditions on these mappings, their definition domains and the spaces in which they…

For $p$ prime and $\ell = \frac{p-1}{2}$, we show that the shapes of pure prime degree number fields lie on one of two $\ell$-dimensional subspaces of the space of shapes, and which of the two subspaces is dictated by whether or not $p$…

For any constant $d$ and parameter $\varepsilon > 0$, we show the existence of (roughly) $1/\varepsilon^d$ orderings on the unit cube $[0,1)^d$, such that any two points $p,q\in [0,1)^d$ that are close together under the Euclidean metric…

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.

The geometry of algebraic curves over finite fields is a rich area of research. In previous work, the authors investigated a particular aspect of the geometry over finite fields of the classical unit circle, namely how the number of…

Semialgebraic splines are functions that are piecewise polynomial with respect to a cell decomposition into sets defined by polynomial inequalities. We study bivariate semialgebraic splines, formulating spaces of semialgebraic splines in…

We investigate the arithmetic of algebraic curves on coarse moduli spaces for special linear rank two local systems on surfaces with fixed boundary traces. We prove a structure theorem for morphisms from the affine line into the moduli…

The well-known analogies between number fields and function fields have led to the transposition of many problems from one domain to the other. In this paper, we will discuss traffic of this sort, in both directions, in the theory of…

This paper gives a theoretical discussion of the orbits and isotropies which arise in a space-time which admits a Lie algebra of Killing vector fields. The submanifold structure of the orbits is explored together with their induced Killing…