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62 works

math.AC — Commutative Algebra

Splitting a Polynomial into Linear Factors after an Injective Ring Extension

Contributed by Darij Grinberg

We show that each univariate polynomial $P = p_0 + p_1 X + \cdots + p_m X^m \in R[X]$ over a commutative ring $R$ can be factored into linear factors over a suitable commutative ring extension $S$ of $R$ . The proof proceeds by universal construction: $S$ is defined as the tensor product $R \otimes_{C_m} B_m$ , where $B_m$ is the polynomial ring $\ZZ[a_1, b_1, a_2, b_2, \ldots, a_m, b_m]$, and where $C_m$ is its subring generated by its ``homogenized elementary symmetric polynomials'' $E_r=\sum_{\substack{I\subseteq [m];\\ |I|=r}} \prod_{i\in I}a_i\prod_{i\notin I}b_i$ for all $0 \leq r \leq m$ . The injectivity of the structure homomorphism $R \to S$ is deduced from a combinatorial study of the diagonal subring of $B_m$ . In the process, a homogeneous variant of the Garsia--Stanton basis is constructed, and some classical properties of symmetric polynomials are recovered.

A mix of human-written and AI-generated textHuman understanding: all parts

Sep 28, 2026PDF ↗View work →

hexagon:2609.00116v1

math.CO — Combinatorics

A Counterexample to a Conjecture on Fused Specht Polynomials

Contributed by Darij Grinberg

Lafay, Peltola and Roussillon conjectured that their realization of simple modules of the fused Hecke algebra by fused Specht polynomials extends from Young diagrams with two columns to Young diagrams of arbitrary shape. We give a counterexample for \[ n=8,\qquad \lambda=(3,2,2,1),\qquad \varsigma=(2,2,2,2). \] More precisely, we exhibit an explicit linear dependence among three fused Specht polynomials indexed by row-strict Young tableaux. The same example also yields an infinite family of counterexamples.

Primarily AI-generated textHuman understanding: some partsSpecht modulesSpecht polynomialsYoung tableaux

Sep 28, 2026PDF ↗View work →

hexagon:2609.00114v4

math.LO — Logic

Under Measurables, HOD_{Ord^{\omega}} is locally arbitrary

Contributed by Elliot Glazer

For the inner model HOD_{Ord^{\omega}}, we derive an analogue for Roguski's classical result that without nontrivial assumptions on V, HOD is an arbitrary model ZFC model. For any sentence \sigma, letting CM denote "class of measurables," the following theories are equiconsistent: (1) ZFC + CM + [HOD_{Ord^\omega} \models (\exists \kappa V_{\kappa} \models \sigma)]; (2) ZFC + GCH + CM + [HOD_{Ord^\omega} \models (SVC \wedge CM \wedge \exists \kappa V_{\kappa} \models \sigma)]; (3) ZF + DC + CM + \exists \kappa V_{\kappa} \models \sigma. Thus, under measurables we have that HOD_{Ord^{\omega}} is "locally arbitrary."

Primarily AI-generated textHuman understanding: some partsSet theoryaxiom of choice

Sep 28, 2026PDF ↗View work →

hexagon:2609.00112v1

math.AT — Algebraic Topology

Swan induction for the finite-local sphere

Contributed by Akhil Mathew

We establish rational Swan induction for ordinary perfect modules over the finite-local sphere $L_n^{p,f}\mathbb S$ . For a finite abelian ambient group, subgroups of $p$ -rank at most $n+1$ suffice, and this bound is sharp. The proof combines cyclic homotopy fixed points in telescopic spectra with the isotropy filtration of a chromatic quotient of finite genuine spectra. The result proves Conjecture~7.22 of Clausen--Mathew--Naumann--Noel for Morava $E$ -theory and gives a new proof of their chromatic upper bound for the algebraic $K$ -theory of $L_n^{p,f}\mathbb S$ -linear categories. The quotient argument also produces finite complexes realizing the induction relations. We formulate the problem of explicit realizations and give two geometric models at the prime two.

Primarily AI-generated textHuman understanding: some partsBurnside ringsSwan inductionchromatic homotopy theory

Sep 27, 2026PDF ↗View work →

hexagon:2609.00111v1

math.AG — Algebraic Geometry

Torsion in $K_1$ and $p$ -adic vanishing cycles

Contributed by Akhil Mathew

For every odd prime $p$ , we construct a regular strictly henselian local ring $A$ whose generic fibre has \'etale cohomology classes in $H^2_{\et}(A[1/p],\mu_p^{\otimes2})$ which are not sums of cup products of degree-one classes. The ring is the strict henselization of a local ring on a finite-type arithmetic scheme. A unit on a principal divisor gives an element of order $p$ in $SK_1(A[1/p])$ , detected by its localization boundary on the special fibre. Adams--Riemann--Roch and the connectivity of the motivic filtration show that this element has nonzero image in $H^3_{\mot}(A[1/p],\Z(2))$. The integral coefficient sequence then gives the cohomological obstruction. Only the eigenvalues $4$ and $8$ of $\psi^2$ are needed.

Primarily AI-generated textHuman understanding: all partsAdams operationsMilnor K-theoryp-adic vanishing cycles

Sep 27, 2026PDF ↗View work →

hexagon:2609.00110v1

math.AG — Algebraic Geometry

Frobenius orbit maps and rank-four group schemes

Contributed by Akhil Mathew

We construct finite free group schemes of rank four and exponent eight as stabilizers in a smooth two-dimensional affine group. We first compute the complete local deformation ring of the invariant space of constants and squares in characteristic two, using six Grassmannian coordinates. For every Artinian specialization, evaluation at the identity has a finite free rank-four stabilizer. Its fourth-power morphism is $(x,y)\mapsto(2bxy,2axy)$ , and its eighth-power morphism is trivial. The fourth power is nonzero on the third-order neighbourhood of the universal deformation ring and on a specialization over $\Z[a,b]/(a^2b-2,a^3,b^3)$ . We give complete formulas for the latter group scheme. Right translation preserves the chosen quadratic space exactly when it contains the square of the scalar character. This identifies $(2a,2b)$ as the common obstruction to fourth-power vanishing, normality of the stabilizer, and two-sidedness in an equivalent cyclic-module construction.

Primarily AI-generated textHuman understanding: some partsHopf algebrasgroup schemes

Sep 27, 2026PDF ↗View work →

hexagon:2609.00102v1

math.NT — Number Theory

Inclusions between p-bounded crystalline loci in dimension two

Contributed by Kalyani Kansal, Brandon Levin, David Savitt

Let p be an odd prime and K/Qp a finite unramified extension of degree f > 1. Let Z(r) be the reduced special fiber of the Emerton-Gee stack of two-dimensional crystalline representations of Hodge type r of the absolute Galois group of K. We study the collection of stacks Z(r) as r varies over p-bounded Hodge types, as a set partially ordered under inclusion. We prove that aside from two degenerate cases, simple inclusions can be classified in terms of three operations on Hodge types, two of which have standard automorphic interpretations. We also prove, with one exception, that inclusions can be detected at the level of an inclusion of closed points (equivalently, semisimple mod p Galois representations).

Primarily human-written textHuman understanding: all parts

Sep 26, 2026PDF ↗View work →

hexagon:2609.00105v1

math.AC — Commutative Algebra

Graded absolute integral closures and Picard groups

Contributed by Akhil Mathew

We deduce from Bhatt's graded Cohen--Macaulay and vanishing results that an integer-graded absolute integral closure of a weighted polynomial ring over $\Z_p$ , modulo $p$ , is free with basis degrees smaller than the sum of the weights. A direct $p$ -adic expansion turns this degree bound into a uniform solution of the cusp relation modulo nilpotents. Consequently every ring $R$ admits a faithfully flat algebra with seminormal reduction, and every invertible $R[T]$ -module becomes free after a faithfully flat extension of $R$ , answering a question of Drinfeld.

Primarily AI-generated textHuman understanding: all partsPicard groupsabsolute integral closureseminormality

Sep 26, 2026PDF ↗View work →

hexagon:2609.00104v1

math.AG — Algebraic Geometry

Derived invariance of the signature and Hodge numbers of fourfolds

Contributed by Benjamin Antieau, Andrei Căldăraru, Ruoxi Li, Akhil Mathew, Joshua Mundinger, Noah Olander, Martin Olsson

A Fourier–Mukai equivalence between smooth proper complex varieties of common even dimension preserves the topological signature. The proof extracts the signature from the symmetrization of the Mukai pairing on even cohomology. Combining this observation with known derived invariants and the Hochschild–Kostant–Rosenberg decomposition shows that derived-equivalent smooth projective fourfolds over any field of characteristic zero have the same Hodge numbers. This argument and text was produced by ChatGPT 5.6 Sol.

Primarily AI-generated textHuman understanding: all partsDerived categoriesFourier–Mukai equivalencefourfolds

Sep 26, 2026PDF ↗View work →

hexagon:2609.00001v1

math.AG — Algebraic Geometry

Strict negativity and non-vanishing of Chen--Larson hypergeometric coefficients

Contributed by Johannes Schmitt

Chen and Larson study tautological classes on the strata of holomorphic abelian differentials, where they predict a vanishing result. Using known tautological relations on the moduli space of curves, they reduce this prediction to the non-vanishing of certain coefficients of a quotient of hypergeometric generating series, treating the residue classes $g\equiv0,2\pmod3$ and $g\equiv1\pmod3$ through two separate series; they verify the non-vanishing by computer for small genus. We prove it in general: in both cases the relevant coefficient is non-vanishing for every genus and every stratum, and in the first case it has, more strongly, a uniform strict sign. Along the way we correct an error in the Chen--Larson derivation of the $g\equiv1\pmod3$ series, where a constant of the underlying relation of Ionel had inadvertently been changed. The paper falls into two parts. Part~I gives the mathematical proofs; Part~II documents the Lean~4 and Mathlib formalization --- whose only non-standard trust assumption is the compiler invoked by \code{native\_decide} for the large finite computations --- together with the long-horizon, multi-model generative-AI process that produced the proofs, exposed false intermediate routes, and uncovered the error in the printed Proposition~5.2 relation noted above.

Primarily AI-generated textHuman understanding: not declaredpower seriestautological classes

Sep 01, 2026PDF ↗View work →