单符号点电荷产生的电场平衡态有限性猜想
The finiteness conjecture for equilibria of electric fields generated by point charges of one sign
- Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas（数学科学研究所，西班牙高等科学研究委员会）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究证明三维空间中有限个单符号点电荷产生的电场仅有有限个平衡点，还给出混合符号点电荷情形下库仑场平衡点数量的定量上界，解答了相关长期猜想。
AI中文摘要：
我们证明，三维空间中有限个单符号点电荷产生的电场仅有有限个平衡点，从而回答了Morse和Cairns在1969年提出、Eremenko在2008年重述、Shapiro在2015年作为猜想提出的问题。更一般地，对于位于R³中不同位置a_i的非零电荷q_i（可能混合符号），我们证明，在S(x):=∑q_i|x−a_i|⁻³不为零的区域内，库仑场至多有2^(N−4)(N−1)(9N²+9N+10)个平衡点。显然，对于单符号电荷，S处处非零。该证明先利用代数几何和相关复曲线上的复分析排除平衡点曲线，再应用Bézout计数得到定量界。已有熟知例子表明，在混合符号情形下，库仑场可在S的零点集内的曲线上消失。
英文摘要：
We prove that the electric field generated in three-dimensional space by finitely many point charges of one sign has only finitely many equilibrium points, thereby answering a 1969 question of Morse and Cairns (restated by Eremenko in 2008 and, as a conjecture, by Shapiro in 2015). More generally, for nonzero charges $q_i$ of possibly mixed signs at distinct sites $\mathbf a_i\in\mathbf{R}^3$, we show that the Coulomb field has at most $2^{N-4}(N-1)(9N^2+9N+10)$ equilibria in the region where $S(\mathbf x):=\sum_iq_i|\mathbf x-\mathbf a_i|^{-3}$ does not vanish. Of course, for charges of one sign, $S$ is nonzero everywhere. The proof rules out curves of equilibria using algebraic geometry and complex analysis on an associated complex curve, and then applies a Bézout count to obtain a quantitative bound. Well-known examples show that, in the mixed-sign case, the Coulomb field can vanish on curves contained in the zero set of $S$.
