Related papers: Congruence Function Fields with Class Number One
We prove a uniqueness theorem for a large class of functional equations in the plane, which resembles in form a classical result of Aczel. It is also shown that functional equations in this class are overdetermined in the sense of Paneah.…
The size function for a number field is an analogue of the dimension of the Riemann-Roch spaces of divisors on an algebraic curve. It was conjectured to attain its maximum at the trivial class of Arakelov divisors. This conjecture was…
We establish that any finite extension of function fields of genus greater than 1 whose relative class group is trivial is Galois and cyclic. This depends on a result from a preceding paper which establishes a finite list of possible Weil…
Rational conformal field theories produce a tower of finite-dimensional representations of surface mapping class groups, acting on the conformal blocks of the theory. We review this formalism. We show that many recent mathematical…
We classify fields having finitely many finite non-commutative (not necessarily central) division algebras over them. In the process, we introduce the notion of anti-closure of a field and also make comments on fields having a linear…
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
We give lower bounds on the number of effective divisors of degree $\leq g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds and…
Let $c_n$ denote the number of nodes at a distance $n$ from the root of a rooted tree. A criterion for proving the rationality and computing the rational generating function of the sequence $\{c_n\}$ is described. This criterion is applied…
We establish bounds on a finite separable extension of function fields in terms of the relative class number, thus reducing the problem of classifying extensions with a fixed relative class number to a finite computation. We also solve the…
We determine all isomorphism classes of hyperfields of a given finite order which can be obtained as quotients of finite fields of sufficiently large order. Using this result, we determine which hyperfields of order at most 4 are quotients…
By solving the Ward identities in a superconformal field theory we find the unique three-point Green's functions composed of chiral superfields for N = 1,2,3,4 supersymmetry. We show that the N=1 four-point function with R-charge equal to…
In this manuscript, we investigate the automorphism group of a maximal function field with the second largest possible genus over finite field of even characteristic, which is called the Abd\'on--Torres function field. As an application, we…
Over a field of characteristic 2, we give a complete classification of quadratic and bilinear forms of dimension 5 that are minimal over the function field of an arbitrary conic. This completes the unique known case due to Faivre concerning…
We obtain several results concerning the concept of isotypic structures. Namely we prove that any field of finite transcendence degree over a prime subfield is defined by types; then we construct isotypic but not isomorphic structures with…
We present two infinite families of APN functions where the degree of the field is divisible by 3 but not 9. Our families contain two already known families as special cases. We also discuss the inequivalence proof (by computation) which…
We prove several results concerning genus numbers of quintic fields: we compute the proportion of quintic fields with genus number one; we prove that a positive proportion of quintic fields have arbitrarily large genus number; and we…
Bennett, Hart, Iosevich, Pakianathan, and Rudnev found an exponent $s<d$ such that any set $E\subset \mathbb{F}_q^d$ with $|E|\gtrsim q^s$ determines $\gtrsim q^{\binom{k+1}{2}}$ congruence classes of $(k+1)$-point configurations for $k\leq…
We classify isomorphism types of unital commutative algebras of rank 7 over an algebraically closed field of characteristic not 2 or 3 completely.
We prove for a large class of fields $F$ that every proper finite extension of $F_{pyth}$, the pythagorean closure of $F$, is not a pythagorean field. This class of fields contains number fields and fields $F$ that are finitely generated of…
We study rational functions over finite fields under PGL-equivalence. We say that $f, g \in \Bbb F_q(X)$ are \emph{equivalent} if there exist $\psi, \phi \in \Bbb F_q(X)$ of degree one such that $g = \psi \circ f \circ \phi$. Most…