arXiv subjects

Elira Shaska

Publications and source records attributed to Elira Shaska.

7 recordsLinked to original sources

arXiv2025

Rational Points and Zeta Functions of Humbert Surfaces with Square Discriminant

This paper examines the arithmetic of the loci $\cL_n$, parameterizing genus 2 curves with $(n, n)$-split Jacobians over finite fields $\F_q$. We compute rational points $|\cL_n(\F_q)|$ over $\F_3$, $\F_9$, $\F_{27}$, $\F_{81}$, and $\F_5$, $\F_{25}$, $\F_{125}$, derive zeta functions $Z(\cL_n, t)$ for $n = 2, 3$. Utilizing these findings, we explore isogeny-based cryptography, introducing an efficient detection method for split Jacobians via explicit equations, enhanced by endomorphism ring analysis and machine learning optimizations. This advances curve selection, security analysis, and protocol design in post-quantum genus 2 systems, addressing efficiency and vulnerabilities across characteristics.

math.NT↗

arXiv2025

Weighted Heights and GIT Heights

We investigate the relationship between Geometric Invariant Theory (GIT) heights and weighted heights, with a focus on their interaction in weighted projective spaces and their application to binary forms. Building on the weighted height framework developed in previous papers, we relate it to Zhang's GIT height via the Veronese map. For a semistable cycle, we show that the GIT height decomposes into the logarithmic weighted height plus an Archimedean correction from the Chow metric.

math.AG↗

arXiv2025

Rational Functions on the Projective Line from a Computational Viewpoint

An explicit invariant-theoretic description of the moduli space $\mathcal{M}_3^1$ of degree-three rational maps on $\mathbb{P}^1$ is developed. A cubic map $\phi$ is represented, up to conjugation, by the pair of binary forms $(f, g) \in V_4 \oplus V_2$ arising from its Clebsch--Gordan decomposition. From this representation one constructs weighted projective invariants $\xi_0, ..., \xi_5$ that embed $\mathcal{M}_3^1$ into $\mathbb{P}^5(2,2,3,3,4,6)$ onto the locus where the gcd of the weights of the non-zero coordinates equals $1$, together with absolute invariants defined as weight-zero rational functions of the $\xi_i$, normalized by an additional invariant $I_6$ of weight $6$. These absolute invariants determine the isomorphism class uniquely. The stratification of $\mathcal{M}_3^1$ is described explicitly by equations in the absolute invariants or polynomial relations among the $\xi_i$. Computational illustrations demonstrate that the resulting invariants provide an effective feature set for automated classification of automorphism groups. The methods suggest natural extensions to higher degrees.

math.AG↗

arXiv2025

Neuro-Symbolic Learning for Galois Groups: Unveiling Probabilistic Trends in Polynomials

This paper presents a neurosymbolic approach to classifying Galois groups of polynomials, integrating classical Galois theory with machine learning to address challenges in algebraic computation. By combining neural networks with symbolic reasoning we develop a model that outperforms purely numerical methods in accuracy and interpretability. Focusing on sextic polynomials with height $\leq 6$, we analyze a database of 53,972 irreducible examples, uncovering novel distributional trends, such as the 20 sextic polynomials with Galois group $C_6$ spanning just seven invariant-defined equivalence classes. These findings offer the first empirical insights into Galois group probabilities under height constraints and lay the groundwork for exploring solvability by radicals. Demonstrating AI's potential to reveal patterns beyond traditional symbolic techniques, this work paves the way for future research in computational algebra, with implications for probabilistic conjectures and higher degree classifications.

cs.LG↗

arXiv2025

Galois groups of polynomials and neurosymbolic networks

This paper introduces a novel approach to understanding Galois theory, one of the foundational areas of algebra, through the lens of machine learning. By analyzing polynomial equations with machine learning techniques, we aim to streamline the process of determining solvability by radicals and explore broader applications within Galois theory. This summary encapsulates the background, methodology, potential applications, and challenges of using data science in Galois theory. More specifically, we design a neurosymbolic network to classify Galois groups and show how this is more efficient than usual neural networks. We discover some very interesting distribution of polynomials for groups not isomorphic to the symmetric groups and alternating groups.

cs.LG↗

arXiv2024

Zgjidhja e ekuacionit t\"e grad\"es s\"e 5-t\"e

An irreducible quintic equation is solvable by radicals if and only if its Galois group is solvable. In this work, we provide necessary and sufficient conditions for solvability, expressed in terms of invariants of the quintic.

math.HO↗

arXiv2024

Machine learning for moduli space of genus two curves and an application to isogeny based cryptography

We use machine learning to study the moduli space of genus two curves, specifically focusing on detecting whether a genus two curve has $(n, n)$-split Jacobian. Based on such techniques, we observe that there are very few rational moduli points with small weighted moduli height and $(n, n)$-split Jacobian for $n=2, 3, 5$. We computational prove that there are only 34 genus two curves (resp. 44 curves) with (2,2)-split Jacobians (resp. (3,3)-split Jacobians) and weighted moduli height $\leq 3$. We discuss different machine learning models for such applications and demonstrate the ability to detect splitting with high accuracy using only the Igusa invariants of the curve. This shows that artificial neural networks and machine learning techniques can be highly reliable for arithmetic questions in the moduli space of genus two curves and may have potential applications in isogeny-based cryptography.

math.AG↗