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Andrew O'Desky

Publications and source records attributed to Andrew O'Desky.

11 recordsLinked to original sources
arXiv2026

On attention heads and bilinear forms

We study the symmetric and antisymmetric parts of bilinear forms in the attention heads of trained large language models. We introduce an orthogonally invariant profile map from real bilinear forms to a three-dimensional simplex and observe that profiles of trained bilinear forms accumulate near profiles of rank-one bilinear forms. We prove that the symmetric part of a bilinear form in an attention head is the sum of a hyperbolic form and a zero form for a Zariski-dense subset of query-key matrices.

cs.LG↗
arXiv2024

Integral points on singular toric varieties and cyclic normal polynomials

We establish a formula for the height zeta function for integral points on a class of projective toric varieties. Our method builds on the harmonic analysis approach of Batyrev--Tschinkel for rational points and is applicable even when the toric variety has cyclic quotient singularities. As an application, we determine the leading term in the asymptotic number of monic integral polynomials of bounded height with linearly independent roots and a given cyclic Galois group.

math.NT↗
arXiv2024

On trace-one generators of abelian cubic fields

Let $K$ be a tamely ramified abelian cubic number field with discriminant $D_K$. We prove that the number of trace-one monic integral polynomials with root field $K$ and height $H$ is equal to the number of ideals in the quadratic field $\mathbb Q(\sqrt{-3})$ with norm $H^2 D_K^{-1/2}$.

math.NT↗
arXiv2023

On monic abelian trace-one cubic polynomials

We compute the asymptotic number of monic trace-one integral polynomials with Galois group $C_3$ and bounded height. For such polynomials we compute a height function coming from toric geometry and introduce a parametrization using the quadratic cyclotomic field $\mathbb Q(\sqrt{-3})$. We also give a formula for the number of polynomials of the form $t^3 -t^2 + at + b \in \mathbb Z[t]$ with Galois group $C_3$ for a fixed integer $a$.

math.NT↗
arXiv2023

The moduli space of $G$-algebras

Let $L$ be a Galois algebra with Galois group $G$ and let $x$ be a normal element of $L$. The moduli space $\mathcal X$ of pairs $(L,x)$ is isomorphic to an open subset of the quotient variety $\mathbb P/G$, where $\mathbb P$ is the projective space of the regular representation of $G$. We provide a formula for the height of any pair $(L,x) \in \mathcal X(\mathbb Q)$ in terms of algebraic invariants of $L$ and $x$ with respect to a natural adelic metric on the anticanonical divisor of $\mathbb P/G$.

math.NT↗
arXiv2022

Derangements and the $p$-adic incomplete gamma function

We introduce a $p$-adic analogue of the incomplete gamma function. We also introduce quantities ($m$-values) associated to a function on natural numbers and prove a new characterization of $p$-adic continuity for functions with $p$-integral $m$-values. Combinatorial interpretations for the integral values of the incomplete gamma function and functions with $m$-values zero or one are obtained, which show that these functions count derangements in generalized symmetric groups and permutations with restricted cycle lengths.

math.NT↗
arXiv2021

Functions with integer-valued divided differences

Let $s_0,s_1,s_2,\ldots$ be a sequence of rational numbers whose $m$th divided difference is integer-valued. We prove that $s_n$ is a polynomial function in $n$ if $s_n \ll θ^n$ for some positive number $θ$ satisfying $θ< e^{1 + \tfrac{1}{2} + \cdots+ \tfrac{1}{m}} -1$.

math.NT↗
arXiv2019

An induction property for prime counting functions

We provide an elementary proof of an asymptotic formula for prime counting functions. As a minor application we give a new reduction of the proof of Chebotarëv's density theorem to the cyclic case.

math.NT↗
arXiv2019

Super Vertex Algebras, Meromorphic Jacobi Forms and Umbral Moonshine

The vector-valued mock modular forms of umbral moonshine may be repackaged into meromorphic Jacobi forms of weight one. In this work we constructively solve two cases of the meromorphic module problem for umbral moonshine. Specifically, for the type A Niemeier root systems with Coxeter numbers seven and thirteen, we construct corresponding bigraded super vertex operator algebras, equip them with actions of the corresponding umbral groups, and verify that the resulting trace functions on canonically twisted modules recover the meromorphic Jacobi forms that are specified by umbral moonshine. We also obtain partial solutions to the meromorphic module problem for the type A Niemeier root systems with Coxeter numbers four and five, by constructing super vertex operator algebras that recover the meromorphic Jacobi forms attached to maximal subgroups of the corresponding umbral groups.

math.RT↗