论文标题
二进制形式基本区域区域的下限
A Lower Bound for the Area of the Fundamental Region of a Binary Form
论文作者
论文摘要
令$ f（x，y）= \ prod \ limits_ {k = 0}^{n -1}（Δ_kx-γ_ky）$$是$ n \ geq 1 $的二进制形式，具有复杂的系数，以$ n $ linear形式写成$ \ \ nathbb c [x，x，y] $。令$ h_f = \ prod \ limits_ {k = 0}^{n -1} \ sqrt {|γ_k|^2 + |^2 + |δ_k|^2} $ $表示$ f $的高度，让$ a_f $ a_f $表示$ f $：$ f $：$ f $： | f（x，y）| \ leq 1 \ right \}。 $$我们证明$ h_f^{2/n} a_f \ geq \ left（2^{1 +（r/n）} \ right）π$，其中$ r $是$ f $ $ f $的真实投影行$ \ \ \ \ \ \ \ \ m m i \ mathbb r \ mathb r \ mathb p^1 $，计数乘数。
Let $$ F(x, y) = \prod\limits_{k = 0}^{n - 1}(δ_kx - γ_ky) $$ 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 $$ h_F = \prod\limits_{k = 0}^{n - 1}\sqrt{|γ_k|^2 + |δ_k|^2} $$ 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 $h_F^{2/n}A_F \geq \left(2^{1 + (r/n)}\right)π$, where $r$ is the number of roots of $F$ on the real projective line $\mathbb R\mathbb P^1$, counting multiplicity.
