Cubic polynomials and sums of two squares
Siddharth Iyer
TL;DR
The paper studies how often an irreducible cubic polynomial $\mathbf{p}$ takes sums of two squares values, by connecting the problem to integer points on the Ch\u00e2telet surface $y^2+z^2=\mathbf{p}(x)$. It develops a two-dimensional unit argument in degree six number fields and a transfer principle in $\mathbb{Z}[\theta]$ to relate $\mathbf{p}(n)\in \square_2$ to representations of $q^2(n-\theta)$ as a sum of two squares in $\mathbb{Z}[\theta]$, yielding the main bound $\#B_{\mathbf{p}}(X)\gg X^{1/3-o(1)}$ under parity conditions $a_2^2-a_1$ even and $a_1a_2-a_0$ odd. The author provides a constructive example achieving $\#\mathcal{R}_{\theta,1}(dX^{1/2},X)\gg X^{1/3-o(1)}$ and discusses potential generalizations to $y^2+nz^2=\mathbf{p}(x)$ and to stronger bounds $\#B_{\mathbf{p}}(X)\gg X^{1/2}$ via special substitutions. The work sheds light on the density of cubic values representable by the quadratic form $x^2+ny^2$ and offers a framework for broader Diophantine questions on Ch\u00e2telet surfaces.
Abstract
We state a lower bound for how often an irreducible monic cubic polynomial is a sum of two squares ($\square_{2}$). To do so, we rely on a two-dimensional unit argument, and degree six number fields. As an example we can show that if $h$ is an odd integer which is not a cube, then $\# \{n: \ n^3+h \in \square_{2}, \ 1 \leq n \leq x \} \gg x^{1/3-o(1)}$. Our arguments may be generalised to generate literature for how often an irreducible monic cubic polynomial is represented in the quadratic form $x^2+ny^2$, provided $n \in \mathbb{N}$.