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

Abstract

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