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

TL;DR

This paper establishes a lower bound on the area of the fundamental region of a binary form based on its height and the number of real roots, contributing to the understanding of geometric properties of polynomial forms.

Contribution

It provides a new lower bound relating the area of the fundamental region of a binary form to its height and real roots count, a novel geometric inequality in algebraic number theory.

Findings

Proves a lower bound: h_F^{2/n}A_F ≥ (2^{1 + (r/n)})π

Relates geometric area to algebraic properties of binary forms

Enhances understanding of the shape and size of fundamental regions

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.