A Lower Bound for the Area of the Fundamental Region of a Binary Form

Abstract

Let $$ F(x, y) = \prod\limits_{k = 0}^{n - 1}(\delta_kx - \gamma_ky) $$ be a binary form of degree $n \geq 1$, with complex coefficients, written as a product of $n$ linear forms in $\mathbb C[x, y]$. Let $$ h_F = \prod\limits_{k = 0}^{n - 1}\sqrt{|\gamma_k|^2 + |\delta_k|^2} $$ denote the height of $F$ and let $A_F$ denote the area of the fundamental region of $F$: $$ \left\{(x, y) \in \mathbb R^2 \colon |F(x, y)| \leq 1\right\}. $$ We prove that $h_F^{2/n}A_F \geq \left(2^{1 + (r/n)}\right)\pi$, where $r$ is the number of roots of $F$ on the real projective line $\mathbb R\mathbb P^1$, counting multiplicity.