发表机构
Nanyang Technological University（南洋理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明了带符号直线猜想，给出带符号点电荷势直线限制的临界点数量的二分结论，还证明了Edelsbrunner–Fillmore–Oliveira的猜想3并扩展其适用范围，同时给出达到\\(2n-1\\)个简单临界点的正n电荷构型。
AI 中文摘要
设\\(\alpha>0\\)，将具有任意实系数的有限逆幂势限制在一条直线上。合并具有相同投影中心和平方高度\\((a,b^2)\\)的项，删除系数和为零的类，令\\(m\\)为剩余类的数量。我们证明以下二分法：若\\(m=0\\)，当且仅当每个类和消失时发生精确抵消，且原定义域的每个普通点都是临界点；若\\(m\geq1\\)，则最多有\\(2m-1\\)个临界点：当所有有效高度为正时，零点按解析重数计数，而当直线上存在有效源时，该断言为一个全局不同点的界。这证明了Gabrielov–Novikov–Shapiro的带符号直线猜想，在其整个范围内成立，且实际上对每个\\(\alpha>0\\)均成立。结构输入是射影配对Haar定理：对于有限\\(\beta>1\\)，全\\(2m\\)维空间\\(\sum L_j/Q_j^\beta\\)（其中\\(Q_j\\)为两两不成比例的正定二元二次型，\\(L_j\\)为任意实线性分子）最多有\\(2m-1\\)个带重数的射影零点。对于正电荷，代入\\(p=2\alpha\\)可证明Edelsbrunner–Fillmore–Oliveira的猜想3在其规定范围\\(p\geq1\\)内成立，并将相同结论扩展到所有\\(p>0\\)。对于每个\\(n\\)，存在显式的正n电荷构型达到\\(2n-1\\)个简单临界点。
英文摘要
Let $α>0$ and restrict a finite inverse-power potential with arbitrary real coefficients to a line. Combine terms having the same projected centre and squared height $\left(a, b^2\right)$, delete classes whose coefficient sum is zero, and let $m$ be the number of remaining classes. We prove the following dichotomy. If $m=0$, exact cancellation occurs if and only if every class sum vanishes, and every ordinary point of the original domain is critical. If $m \geq 1$, there are at most $2 m-1$ critical points: when all effective heights are positive the zeros are counted with analytic multiplicity, while in the presence of effective sources on the line the assertion is one global distinct-point bound. This proves the signed line conjecture of Gabrielov-Novikov-Shapiro, valid throughout their range and in fact for every $α>0$. The structural input is a projective paired Haar theorem: for finite $β>1$, the full $2 m$-dimensional space $\sum L_j / Q_j^β$, with pairwise nonproportional positive-definite binary quadratics and arbitrary real linear numerators, has at most $2 m-1$ projective zeros counted with multiplicity. For positive charges, the substitution $p=2 α$ proves Conjecture 3 of Edelsbrunner-Fillmore-Oliveira throughout its stated range $p \geq 1$ and extends the same conclusion to every $p>0$. For every $n$, an explicit positive $n$-charge configuration attains $2 n-1$ simple critical points.
