The arithmetic of critical values I: equicritical quartic polynomials

TL;DR

This paper develops a framework for studying the arithmetic of critical values of degree 4 polynomials over algebraically closed fields, revealing connections to elliptic curves and classifying pairs with identical critical values over number fields.

Contribution

It introduces a rigorous approach to the arithmetic of critical values for quartic polynomials and classifies pairs with the same critical values over number fields.

Findings

Established a framework for the arithmetic study of critical values of quartic polynomials.

Connected the critical values of these polynomials to the arithmetic of elliptic curves.

Provided a complete classification of quartic polynomial pairs with identical critical values over number fields.

Abstract

A polynomial $f$ of degree $d$ and coefficients in an algebraically closed field $k$ defines a morphism $f:\mathbb{P}^1_k\longrightarrow\mathbb{P}^1_k$ which, if char$(k)\nmid d$, is unramified outside a finite set of points in the image: the critical values of $f$. In this work we establish a rigorous framework for the study of their arithmetic, which we carry out for $d=4$ and $k=\overline{\mathbb{Q}}$, uncovering a connection to the arithmetic of elliptic curves. Recent progress in the theory of Weyl sums has sparked some interest in finding pairs of polynomials having the same critical values for "nontrivial" reasons: building on our analysis, we provide a complete classification of such pairs in the case of quartics over number fields.