Cubic Polynomials and Sums of Two Squares

TL;DR

This paper investigates how often irreducible monic cubic polynomials with negative discriminant can be expressed as sums of two squares, providing quantitative bounds and extending to related quadratic forms.

Contribution

It establishes lower bounds for the frequency of such representations and introduces methods involving degree six number fields and unit arguments.

Findings

Shows that for certain congruences, the count of n with n^3+h as a sum of two squares grows at least as x^{1/3-o(1)}.

Provides a quantitative answer to Grechuk's question on the infinitude of such values.

Suggests generalizations to other quadratic forms like x^2 + ny^2.

Abstract

We establish a lower bound for the frequency with which an irreducible monic cubic polynomial with negative discriminant can be expressed as a sum of two squares ($\square_{2}$). This provides a quantitative answer to a question posed by Grechuk (2021) concerning the infinitude of such values. Our proof relies on a two-dimensional unit argument and the arithmetic of degree six number fields. For example, we show that if $h \equiv 2 \pmod{4}$, then \begin{align*} \# \{n : n^3+h \in \square_{2}, \ 1 \leq n \leq x \} \gg x^{1/3-o(1)}. \end{align*} These arguments may be generalised to study the representation of irreducible monic cubic polynomials by the quadratic form $x^2+ny^2$, where $n \in \mathbb{N}$.