arXiv subjects

Kelly McKinnie

Publications and source records attributed to Kelly McKinnie.

At least 19 recordsLinked to original sources

arXiv2026

Density of Visible Lattice Points on Hyperplanes and their Intersections

A lattice point $\vec x=(x_1,\dots,x_n)\in\mathbb Z^{n}$ is said to be visible if the line segment between $\vec x$ and the origin contains no other lattice point. In this paper, we compute the asymptotic density of visible lattice points on hyperplanes and their intersections. In particular, we show that the hyperplane $\vec a \cdot \vec x = b$ in $\mathbb R^{n}$ has visible point density $J_{n-1}(b)/b^{n-1}$ where $J$ is the Jordan totient function. We extend this basic result to find the density of visible points on the intersection of hyperplanes and to the density of $k$-th power free points. Finally, for a fixed dimension $n$, we consider the closure of the set of all possible densities that occur.

math.NT↗

arXiv2025

Essential Dimension of Central Simple Algebras when the Characteristic is Bad

This is a survey of the existing literature, the state of the art, and a few minor new results and open questions regarding the essential dimension of central simple algebras and finite sequences of such algebras over fields whose characteristic divides the degree of the algebras under discussion. Upper and lower bounds as well as a few precise evaluations of this dimension are included.

math.RA↗

arXiv2024

Classes in $\mathrm H_{p^m}^{n+1}(F)$ of lower exponent

Let $F$ be a field of characteristic $p>0$. We prove that if a symbol $A=\omega \otimes \beta_1 \otimes \dots \otimes \beta_n$ in $H_{p^m}^{n+1}(F)$ is of exponent dividing $p^{m-1}$, then its symbol length in $H_{p^{m-1}}^{n+1}(F)$ is at most $p^n$. In the case $n=2$ we also prove that if $A= \omega_1\otimes \beta_1+\cdots+\omega_r\otimes \beta_r$ in $H_{p^{m}}^2(F)$ satisfies $\exp(A)|p^{m-1}$, then the symbol length of $A$ in $H_{p^{m-1}}^2(F)$ is at most $p^r+r-1$. We conclude by looking at the case $p=2$ and proving that if $A$ is a sum of two symbols in $H_{2^m}^{n+1}(F)$ and $\exp A |2^{m-1}$, then the symbol length of $A$ in $H_{2^{m-1}}^{n+1}(F)$ is at most $(2n+1)2^n$. Our results use norm conditions in characteristic $p$ in the same manner as Matrzi in his paper ``On the symbol length of symbols''.

math.RA↗

arXiv2024

A retrospective analysis of Montana's 2020 congressional redistricting map

The 2020 decennial census data resulted in an increase from one to two congressional representatives in the state of Montana. The state underwent its redistricting process in 2021 in time for the November 2022 congressional elections, carving the state into two districts. This paper analyzes the redistricting process and compares the adopted congressional map to the space of all other possible maps. In particular, we look at the population deviation, compactness and political outcomes of these maps. We also consider how well two popular sampling techniques, that sample from the space of possible maps, approximate the true distributions of these measures.

cs.CY↗

arXiv2023

Biquaternion Algebras, Chain Lemma and Symbol Length

In this note, we present a chain lemma for biquaternion algebras over fields of characteristic 2 in the style of the equivalent chain lemma by Sivatski in characteristic not 2, and conclude a bound on the symbol length of classes in ${_{2^n}Br}(F)$ whose symbol length in ${_{2^{n+1}}Br}(F)$ is at most 4.

math.RA↗

arXiv2022

Common Splitting Fields of Symbol Algebras

We study the common splitting fields of symbol algebras of degree $p^m$ over fields $F$ of $\operatorname{char}(F)=p$. We first show that if any finite number of such algebras share a degree $p^m$ simple purely inseparable splitting field, then they share a cyclic splitting field of the same degree. As a consequence, we conclude that every finite number of symbol algebras of degrees $p^{m_0},\dots,p^{m_t}$ share a cyclic splitting field of degree $p^{m_0+\dots+m_t}$. This generalization recovers the known fact that every tensor product of symbol algebras is a symbol algebra. We apply a result of Tignol's to bound the symbol length of classes in $\operatorname{Br}_{p^m}(F)$ whose symbol length when embedded into $\operatorname{Br}_{p^{m+1}}(F)$ is 2 for $p\in \{2,3\}$. We also study similar situations in other Kato-Milne cohomology groups, where the necessary norm conditions for splitting exist.

math.RA↗

arXiv2021

Asymptotic Brauer $p$-Dimension

We define and compute $\operatorname{ABrd}_p(F)$, the asymptotic Brauer $p$-dimension of a field $F$, in cases where $F$ is a rational function field or Laurent series field. $\operatorname{ABrd}_p(F)$ is defined like the Brauer $p$-dimension except it considers finite sets of Brauer classes instead of single classes. Our main result shows that for fields $F_0(α_1,\dots,α_n)$ and $F_0 (\!( α_1)\!) \dots(\!(α_n)\!)$ where $F_0$ is a perfect field of characteristic $p>0$ when $n \geq 2$ the asymptotic Brauer $p$-dimension is $n$. We also show that it is $n-1$ when $F=F_0 (\!( α_1)\!) \dots(\!(α_n)\!)$ and $F_0$ is algebraically closed of characteristic not $p$. We conclude the paper with examples of pairs of cyclic algebras of odd prime degree $p$ over a field $F$ for which $\operatorname{Brd}_p(F)=2$ that share no maximal subfields despite their tensor product being non-division.

math.RA↗

arXiv2020

Farey Recursive Functions

This paper introduces Farey Recursive Functions and investigates their basic properties. Farey Recursive Functions are a special type of recursive function from the rationals to a commutative ring. The recursion of these functions is organized by the Farey graph. They arise naturally in the study of 2-bridge knots and links.

math.GT↗

arXiv2020

Essential Dimension, Symbol Length and $p$-rank

We prove that the essential dimension of central simple algebras of degree $p^{\ell m}$ and exponent $p^m$ over fields $F$ containing a base-field $k$ of characteristic $p$ is at least $\ell+1$ when $k$ is perfect. We do this by observing that the $p$-rank of $F$ bounds the symbol length in $\operatorname{Br}_{p^m}(F)$ and that there exist indecomposable $p$-algebras of degree $p^{\ell m}$ and exponent $p^m$. We also prove that the symbol length of the Milne-Kato cohomology group $\operatorname H^{n+1}_{p^m}(F)$ is bounded from above by $\binom rn$ where $r$ is the $p$-rank of the field, and provide upper and lower bounds for the essential dimension of Brauer classes of a given symbol length.

math.RA↗

arXiv2018

Kato-Milne Cohomology and Polynomial Forms

Given a prime number $p$, a field $F$ with $\operatorname{char}(F)=p$ and a positive integer $n$, we study the class-preserving modifications of Kato-Milne classes of decomposable differential forms. These modifications demonstrate a natural connection between differential forms and $p$-regular forms. A $p$-regular form is defined to be a homogeneous polynomial form of degree $p$ for which there is no nonzero point where all the order $p-1$ partial derivatives vanish simultaneously. We define a $\widetilde C_{p,m}$ field to be a field over which every $p$-regular form of dimension greater than $p^m$ is isotropic. The main results are that for a $\widetilde C_{p,m}$ field $F$, the symbol length of $H_p^2(F)$ is bounded from above by $p^{m-1}-1$ and for any $n \geq \lceil (m-1) \log_2(p) \rceil+1$, $H_p^{n+1}(F)=0$.

math.RA↗

arXiv2017

The $u^n$-invariant and the Symbol Length of $H_2^n(F)$

Given a field $F$ of $\operatorname{char}(F)=2$, we define $u^n(F)$ to be the maximal dimension of an anisotropic form in $I_q^n F$. For $n=1$ it recaptures the definition of $u(F)$. We study the relations between this value and the symbol length of $H_2^n(F)$, denoted by $sl_2^n(F)$. We show for any $n \geq 2$ that if $2^n \leq u^n(F) \leq u^2(F) < \infty$ then $sl_2^n(F) \leq \prod_{i=2}^n (\frac{u^i(F)}{2}+1-2^{i-1})$. As a result, if $u(F)$ is finite then $sl_2^n(F)$ is finite for any $n$, a fact which was previously proven when $\operatorname{char}(F) \neq 2$ by Saltman and Krashen. We also show that if $sl_2^n(F)=1$ then $u^n(F)$ is either $2^n$ or $2^{n+1}$.

math.AC↗

arXiv2017

Essential Dimension of Generic Symbols in Characteristic p

In this article the $p$-essential dimension of generic symbols over fields of characteristic $p$ is studied. In particular, the $p$-essential dimension of the length $\ell$ generic $p$-symbol of degree $n+1$ is bounded below by $n+\ell$ when the base field is algebraically closed of characteristic $p$. The proof uses new techniques for working with residues in Milne-Kato $p$-cohomology and builds on work of Babic and Chernousov in the Witt group in characteristic 2. Two corollaries on $p$-symbol algebras (i.e, degree 2 symbols) result from this work. The generic $p$-symbol algebra of length $\ell$ is shown to have $p$-essential dimension equal to $\ell+1$ as a $p$-torsion Brauer class. The second is a lower bound of $\ell+1$ on the $p$-essential dimension of the functor $\mathrm{Alg}_{p^\ell,p}$. Roughly speaking this says that you will need at least $\ell+1$ independent parameters to be able to specify any given algebra of degree $p^{\ell}$ and exponent $p$ over a field of characteristic $p$ and improves on the previously established lower bound of 3.

math.RA↗

arXiv2015

Brauer groups on K3 surfaces and arithmetic applications

For a prime $p$, we study subgroups of order p of the Brauer group Br(S) of a general complex polarized K3 surface of degree 2d, generalizing earlier work of van Geemen. These groups correspond to sublattices of index p of the transcendental lattice T_S of S; we classify these lattices up to isomorphism using Nikulin's discriminant form technique. We then study geometric realizations of p-torsion Brauer elements as Brauer-Severi varieties in a few cases via projective duality. We use one of these constructions for an arithmetic application, giving new kinds of counter-examples to weak approximation on K3 surfaces of degree two.

math.AG↗

arXiv2013

Cyclic Length in the Tame Brauer Group of the Function Field of a p-Adic Curve

Let $F$ be the function field of a smooth curve over the $p$-adic number field $\Q_p$. We show that for each prime-to-$p$ number $n$ the $n$-torsion subgroup $\H^2(F,μ_n)={}_n\Br(F)$ is generated by $\Z/n$-cyclic classes; in fact the $\Z/n$-length is equal to two. It follows that the Brauer dimension of $F$ is two (first proved in \cite{Sa97}), and any $F$-division algebra of period $n$ and index $n^2$ is decomposable.

math.RA↗

arXiv2010

Indecomposable and noncrossed product division algebras over function fields of smooth p-adic curves

We construct indecomposable and noncrossed product division algebras over function fields of smooth curves X over Z_p. This is done by defining an index preserving morphism s:Br(\hat K(X))' -> Br(K(X))' which splits res:Br(K(X)) -> Br(\hat K(X)), where \hat K(X) is the completion of K(X) at the special fiber, and using it to lift indecomposable and noncrossed product division algebras over \hat K(X).

math.RA↗

arXiv2010

Degeneracy and decomposability in abelian crossed products

In this paper we study the relationship between degeneracy and decomposability in abelian crossed products. In particular we construct an indecomposable abelian crossed product division algebra of exponent $p$ and index $p^2$ for $p$ an odd prime. The algebra we construct is generic in the sense of Amitsur and Saltman and has the property that its underlying abelian crossed product is a decomposable division algebra defined by a non-degenerate matrix. This algebra gives an example of an indecomposable generic abelian crossed product which is shown to be indecomposable without using torsion in the Chow group of the corresponding Severi-Brauer variety as was needed in [Karpenko, Codimension 2 cycles on Severi-Brauer varieites (1998)] and [McKinnie, Indecomposable $p$-algebras and Galois subfields in generic abelian crossed products (2008)]. It also gives an example of a Brauer class which is in Tignol's Dec group with respect to one abelian maximal subfield, but not in the Dec group with respect to another.

math.RA↗

arXiv2010

Distinguishing division algebras by finite splitting fields

This paper is concerned with the problem of determining the number of division algebras which share the same collection of finite splitting fields. As a corollary we are able to determine when two central division algebras may be distinguished by their finite splitting fields over certain fields.

math.RA↗

arXiv2007

Indecomposable p-algebras and Galois subfields in generic abelian crossed products

Let F be a Henselian valued field with char(F) = p and D a semi-ramified, "not strongly degenerate" p-algebra. We show that all Galois subfields of D are inertial. Using this as a tool we study generic abelian crossed product p-algebras, proving among other things that the noncyclic generic abelian crossed product p-algebras defined by non-degenerate matrices are indecomposable p-algebras. To construct examples of these indecomposable p-algebras with exponent p and large index we study the relationship between degeneracy in matrices defining abelian crossed products and torsion in CH^2 of Severi-Brauer varieties.

math.RA↗