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

2022/12/16 by Jason Fang, Fang, Jason, Anton Mosunov +1
Mathematics · #11D75 #FOS: Mathematics #Mathematical Approximation and Integration #Mathematics and Applications #Number Theory (math.NT) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2212.08752

openalex publication_date 2022/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let F(x, y) = ∏_{k = 0}^{n - 1}(δ_kx - γ_ky) be a binary form of degree n ≥ 1, with complex coefficients, written as a product of n linear forms in \mathbb C[x, y]. Let h_F = ∏_{k = 0}^{n - 1}√(|γ_k|² + |δ_k|²) denote the height of F and let A_F denote the area of the fundamental region of F: \(x, y) ∈ \mathbb R² \colon |F(x, y)| ≤ 1\. We prove that h_F^2/nA_F ≥ (2^{1 + (r/n)})π, where r is the number of roots of F on the real projective line \mathbb R\mathbb P¹, counting multiplicity.

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