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Preprint Sep 2026

Schrijver-Delsarte rigidity in association schemes and undecidability of quantum graph homomorphism

We prove RE-completeness of the quantum homomorphism problem parameterised by families of graphs derived from the classic metric association schemes. These include Kneser graphs, $q$-Kneser graphs, and the complements of Johnson, Grassmann, and Hamming graphs. Our proof develops a spectral method for establishing non-c...

Lorenzo Ciardo, Iris Hebbeker, Gideo Joubert et al. · 0 citations
Open access Oct 2026

A note on toric ideals of graphs and Knutson-Miller-Yong Decompositions

We use a Gröbner basis technique first introduced by Knutson, Miller and Yong to study the interplay between properties of a graph [Formula: see text] and algebraic properties of the toric ideal that it defines. We first recover a well-known height formula for the toric ideal of a graph [Formula: see text] and demonstr...

Sergio Da Silva, Emma Naguit, Jenna Rajchgot · 0 citations
Preprint Sep 2026

Pierce-Birkhoff conjecture is false

We provide a counterexample to the Pierce-Birkhoff conjecture: a continuous function that is piecewise quadratic on a finite collection of semialgebraic sets that partition $\mathbb{R}^n$ but that cannot be expressed as a finite lattice combination of polynomials. The counterexample was found with the assistance of our...

Ze-Hua Lai, Lek-Heng Lim, Jun-Yu Ren · 0 citations
Preprint Sep 2026

The Borovik-Cherlin conjecture holds in ACF

We show that every faithful, transitive, and generically $(n+2)$-transitive action of a connected group $G$ on an irreducible variety $X$ of dimension $n>0$, all defined over an algebraically closed field $F$, is isomorphic to the natural action of the projective linear group $PGL_{n+1}(F)$ on the projective space $\ma...

Ulla Karhumäki, Nicholas Ramsey · 0 citations
Jul 2025

On the uniform positivity of 𝐹-signature under reduction modulo 𝑝

Abstract Carvajal-Rojas, Schwede and Tucker asked whether the mod 𝑝 reductions of a complex klt type singularity have uniformly positive 𝐹-signature for all but finitely many primes 𝑝. In this paper, we give an affirmative answer to this conjecture for pure subrings of regular local rings, including reductive quotient...

S. Takagi, Tatsuki Yamaguchi · 5 citations · ⚡1
Preprint Oct 2026

Ranks of operators and pureness of C*-algebras

For simple, nonelementary C*-algebras, we show uniform density of functions realized as ranks of operators. It follows that strict comparison by traces forces almost divisibility, and hence pureness. As an application, we show that strict comparison implies $\mathcal{Z}$-stability for separable, simple, nonelementary C...

Hannes Thiel · 0 citations

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