arXiv subjects

Ariana Chin

Publications and source records attributed to Ariana Chin.

4 recordsLinked to original sources

arXiv2026

Half-periodicity of Zamolodchikov periodic cluster algebras

In 2007, Fomin and Zelevinsky introduced the bipartite belt, a sequence of bipartite mutations whose exchange relations form a discrete dynamical system. Periodicity of this system is known as Zamolodchikov periodicity. In our previous work we have classified all Zamolodchikov periodic cluster algebras, but behavior halfway through the period was still unknown. This so-called half-periodicity was conjectured by Kuniba--Nakanishi--Suzuki for $Y$-systems of finite type Cartan matrices, and was proved by Inoue--Iyama--Keller--Kuniba--Nakanishi for tensor products of two simply-laced Dynkin diagrams. In this paper, we prove that for any Zamolodchikov periodic cluster algebra, the form at the half-period is a permutation of the cluster variables of order at most two.

math.CO↗

arXiv2025

Classification of Zamolodchikov periodic cluster algebras

Zamolodchikov periodicity is a property of certain discrete dynamical systems and was one of the primary motivations for the creation of cluster algebras. It was first observed by Zamolodchikov in his study of thermodynamic Bethe ansatz, initially for simply-laced Dynkin diagrams. It was proved by Keller to hold for tensor products of two Dynkin diagrams, and further shown by Galashin and Pylyavskyy to hold for pairs of commuting simply-laced Cartan matrices of finite type, which Stembridge classified in his study of admissible $W$-cells. We prove that the Zamolodchikov periodic cluster algebras are in bijection with pairs of commuting (not necessarily reduced or simply-laced) Cartan matrices of finite type. We fully classify all such pairs into 29 infinite families and 14 exceptional types in addition to the 6 infinite families and 11 exceptional types in Stembridge's classification, and show that all of these families can be derived from simply-laced types through two operations preserving Zamolodchikov periodicity, folding and taking transpose. Our work holds connections to Kazhdan--Lusztig theory, and our main theorem helps classify all nonnegative $W$-cells for products of two dihedral groups, $W = I_2(p)\times I_2(q)$.

math.CO↗

arXiv2020

The Polynomial Learning With Errors Problem and the Smearing Condition

As quantum computing advances rapidly, guaranteeing the security of cryptographic protocols resistant to quantum attacks is paramount. Some leading candidate cryptosystems use the Learning with Errors (LWE) problem, attractive for its simplicity and hardness guaranteed by reductions from hard computational lattice problems. Its algebraic variants, Ring-Learning with Errors (RLWE) and Polynomial Learning with Errors (PLWE), gain in efficiency over standard LWE, but their security remains to be thoroughly investigated. In this work, we consider the "smearing" condition, a condition for attacks on PLWE and RLWE introduced in [6]. We expand upon some questions about smearing posed by Elias et al. in [6] and show how smearing is related to the Coupon Collector's Problem Furthermore, we develop some practical algorithms for calculating probabilities related to smearing. Finally, we present a smearing-based attack on PLWE, and demonstrate its effectiveness.

cs.IT↗

arXiv2020

The Ring Learning With Errors Problem: Spectral Distortion

We answer a question posed by Y. Elias et al. in [8] about possible spectral distortions of algebraic numbers. We provide a closed form for the spectral distortion of certain classes of cyclotomic polynomials. Moreover, we present a bound on the spectral distortion of cyclotomic polynomials.

math.NT↗