arXiv subjects

Adam Chapman

Publications and source records attributed to Adam Chapman.

At least 19 recordsLinked to original sources

arXiv2026

Points with Commuting Coordinates over Division Rings

We investigate the properties of multivariate polynomials evaluated at points with commuting coordinates over division rings and octonion algebras. Given a division ring $D$, this set of points is denoted by $D_c^n$, and in the special case of $D=\mathbb{H}$, Alon and Paran showed that its points correspond to the maximal left ideals of $\mathbb{H}[x_1,\dots,x_n]$. Here we show that if a polynomial $f$ vanishes at $\vec{a} \in D_c^n$, then any left multiple $gf$ also vanishes at $\vec{a}$. Consequently, over a central division algebra or an octonion algebra, any root of $f$ in $D_c^n$ is also a root of its (reduced) norm. We apply these evaluation properties to discrete algebraic dynamics, proving that if a point in $D_c^n$ is a fixed point of an $n$-tuple $T=(f_1,\dots,f_n)$ of polynomials in $n$ variables, then it is a fixed point of $T^{\circ m}$ for any positive integer $m$.

math.RA↗

arXiv2026

General Polynomials and Eigenvalues Over Cayley--Dickson Algebras

In this paper, we provide an explicit method for determining the zero set of monic quadratic general polynomials over Cayley--Dickson algebras with any base field of characteristic not equal to $2$. For the special case of locally-complex Cayley--Dickson algebras over the reals, we prove that every such polynomial always has a root. We prove the existence of right eigenvalues for any $2 \times 2$ matrix over Cayley's Octonions (the real octonion division algebra), and our root-finding method allows the computation of some of the right eigenvalues.

math.RA↗

arXiv2026

Sums of two symbols in $K_2(F)/2K_2(F)$ in characteristic two

In this paper, study sums $A=\{a,b\}_2+\{c,d\}_2$ of two symbols in $K_2(F)/2K_2(F)$ when $\operatorname{char}(F)=2$. We first prove a chain lemma that connects $A$ to $B=\{\alpha,\beta\}_2+\{\gamma,\delta\}_2$ by a finite sequence of small steps when $A \equiv B$. We use this lemma to prove that $\{a,b,c,d\}_2 \in K_4(F)/2K_4(F)$ is a well-defined invariant of $A$, and that this invariant is trivial if and only if $A$ is congruent to a single symbol in $K_2(F)/4K_2(F)$. We also bound the symbol length of $C$ in $K_2(F)/2^m K_2(F)$ from above when $C$ is the sum of up to four symbols in $K_2(F)/2^{m+1}K_2(F)$.

math.KT↗

arXiv2026

Central Products of Cayley-Dickson Loops

This paper studies the triviality of commutators in central products of Cayley-Dickson loops. Two immediate outcomes of this study are (1) the construction of a sequence of non-commutative di-associative loops in which the probability that a random commutator is trivial approaches 1, and (2) an easy proof that if two central products of $n$-fold Cayley-Dickson loops are isomorphic for $n\geq 3$, then the loops in the first product are term-wise isomorphic to the loops in the second product.

math.RA↗

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↗

arXiv2025

The Cyclicity of Tensor Products of Cyclic $p$-Algebras

We revisit the famous theorem of Albert's on the cyclicity of tensor products of cyclic $p$-algebras. In the case of tensor products of cyclic $p$-algebras of prime degree, we provide an explicit computation of the resulting cyclic algebra in symbol algebra terms.

math.RA↗

arXiv2025

Recurrence relations over division algebras

We generalize the solution of linear recurrence relations from fields to central division algebras, adapting the standard tools of companion matrices and characteristic polynomials to the non-commutative setting. We then solve linear recurrences of order 2 over octonion division algebras.

math.RA↗

arXiv2025

Cyclic Division Algebras of Odd Prime Degree are never Amitsur-Small

A division ring $D$ is Amitsur-Small if for every $n$ and every maximal left ideal $I$ in $D[x_1,\dots,x_n]$, $I \cap D[x_1,\dots,x_{n-1}]$ is maximal in $D[x_1,\dots,x_{n-1}]$. The goal of this note is to prove that cyclic division algebras of odd prime degree over their center are never Amitsur-Small.

math.RA↗

arXiv2025

Reasoning about Medical Triage Optimization with Logic Programming

We present a logic programming framework that orchestrates multiple variants of an optimization problem and reasons about their results to support high-stakes medical decision-making. The logic programming layer coordinates the construction and evaluation of multiple optimization formulations, translating solutions into logical facts that support further symbolic reasoning and ensure efficient resource allocation -- specifically targeting the "right patient, right platform, right escort, right time, right destination" principle. This capability is integrated into GuardianTwin, a decision support system for Forward Medical Evacuation (MEDEVAC), where rapid and explainable resource allocation is critical. Through a series of experiments, our framework demonstrates an average reduction in casualties by 35.75% compared to standard baselines. Additionally, we explore how users engage with the system via an intuitive interface that delivers explainable insights, ultimately enhancing decision-making in critical situations. This work demonstrates how logic programming can serve as a foundation for modular, interpretable, and operationally effective optimization in mission-critical domains.

cs.LO↗

arXiv2025

Essential dimension of sequences of quadratic Pfister forms

We study the essential dimension of the set of isometry classes of $m$-tuples $(\varphi_1,...,\varphi_m)$ of quadratic $n$-fold Pfister forms over a field $F$ such that the Witt class of $\varphi_1 \perp \ldots \perp \varphi_m$ lies in $I_q^{n+1}F$. We show that the essential dimension is equal to $n+1$, when $m=3$, and is either $4$ or $5$, when $n=\text{char} F=2$, $m=4$.

math.NT↗

arXiv2025

5-dimensional minimal quadratic and bilinear forms over function fields of conics

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 the classification of minimal quadratic forms of dimension 5 and type (2,1) over function fields of nonsingular conics.

math.AC↗

arXiv2024

Loops with involution and the Cayley-Dickson doubling process

We develop a theory of loops with involution. On this basis we define a Cayley-Dickson doubling on loops, and use it to investigate the lattice of varieties of loops with involution, focusing on properties that remain valid in the Cayley-Dickson double. Specializing to central-by-abelian loops with elementary abelian $2$-group quotients, we find conditions under which one can characterize the automorphism groups of iterated Cayley-Dickson doubles. A key result is a corrected proof that for $n>3$, the automorphism group of the Cayley-Dickson loop $Q_n$ is $\text{GL}_3(\mathbb{F}_2) \times \{\pm 1\}^{n-3}$.

math.CO↗

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

On the geometry of zero sets of central quaternionic polynomials II

Following the work of the first and last authors [2], we further analyze the structure of a zero set of a left ideal in the ring of central polynomials over the quaternion algebra H. We describe the "algebraic hull" of a point in H^n and prove it is a product of spheres. Using this description we give a new proof to a conjecture of Gori, Sarfatti and Vlacci. We also show that the main result of [2] does not extend to general division algebras.

math.RA↗

arXiv2024

Mixed multiquadratic splitting fields

We study mixed multiquadratic field extensions as splitting fields for central simple algebras of exponent $2$ in characteristic $2$. As an application, we provide examples of nonexcellent mixed biquadratic field extensions.

math.NT↗

arXiv2024

Invariant for Sets of Pfister Forms

We associate an $(n_1+\dots+n_t-k(t-1))$-fold Pfister form to any $t$-tuple of $k$-linked Pfister forms of dimensions $2^{n_1},\dots,2^{n_t}$, and prove its invariance under the different symbol presentations of the forms with a common $k$-fold sub-symbol. We then show that it vanishes when the forms are actually $(k+1)$-linked or when the characteristic is 2 and the forms are inseparably $k$-linked. We study whether the converse statements hold or not.

math.KT↗

arXiv2024

Spaces with Vanishing Characteristic Coefficients

We prove that the maximal dimension of a subspace $V$ of the generic tensor product of $m$ symbol algebras of prime degree $p$ with $\operatorname{Tr}(v^{p-1})=0$ for all $v\in V$ is $\frac{p^{2m}-1}{p-1}$. The same upper bound is thus obtained for $V$ with $\operatorname{Tr}(v)=\operatorname{Tr}(v^2)=\dots=\operatorname{Tr}(v^{p-1})=0$ for all $v \in V$. We make use of the fact that for any subset $S$ of $\underbrace{\mathbb{F}_p \times \dots \times \mathbb{F}_p}_{n \ \text{times}}$ of $|S| > \frac{p^{n}-1}{p-1}$, for all $u\in V$ there exist $v,w\in S$ and $k\in [\![0,p-1]\!]$ such that $kv+(p-1-k)w=u$.

math.RA↗

arXiv2024

Linkage and Essential $p$-Dimension

We prove that two cyclically linked $p$-algebras of prime degree become inseparably linked under a prime to $p$ extension if and only if the essential $p$-dimension of the pair is 2. We conclude that the essential $p$-dimension of pairs of cyclically linked $p$-algebras is 3 by constructing an example of a pair that does not become inseparably linked under any prime to $p$ extension.

math.RA↗