Cubic Polynomials and Sums of Two Squares

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 (${\mathrm{\square }}_2$). 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 $h\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}4)$, 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 $x^2+n{y}^2$, where $n\in \mathbb{N}$.