We examine the relative equilibria of the four vortex problem where three vortices have equal strength, that is \( \varGamma _1 = \varGamma _2 = \varGamma _3 = 1 \) and \( \varGamma _4 = m \), where m is a parameter. We study the problem in the case of the classical logarithmic vortex Hamiltonian and in the case the Hamiltonian is a homogeneous function of degree \( \alpha \). First we study the bifurcations emanating from the equilateral triangle configuration with the fourth vortex in the barycenter. Then, in the vortex case, we show that all the convex configurations contain a line of symmetry. We also give a precise count of the number of concave asymmetric configurations. Our techniques involve a combination of analysis and computational algebraic geometry.

中文翻译：

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涡旋和一般同质势的相对平衡分叉

我们检查四个涡旋问题的相对平衡，其中三个涡旋具有相同的强度，即\（\ varGamma _1 = \ varGamma _2 = \ varGamma _3 = 1 \）和\（\ varGamma _4 = m \），其中m为参数。我们研究经典对数涡旋哈密顿量和哈密顿量为\（\ alpha \）的齐次函数的问题。首先，我们研究由重心中具有第四个涡旋的等边三角形构型产生的分叉。然后，在涡旋情况下，我们表明所有凸形结构都包含一条对称线。我们还给出了凹形不对称结构数量的精确计数。我们的技术涉及分析和计算代数几何的组合。