关于演化系统平衡点最大个数的Feldman-Karlin猜想的证明
Proof of the Feldman-Karlin Conjecture on the Maximum Number of Equilibria in an Evolutionary System
摘要：
费尔德曼和卡林指出，在n个可能的单倍型中，生存力选择和重组的确定性模型的孤立不动点的数量上限为2^n - 1。这里提供了一个证明。由Lyubich等人（2001）使用Bezout定理（1779）获得的3^{n-1}的上限在这里通过将三阶多项式降低到二阶的表示的改变而降低到2^n。使用系统的齐次表示，得到了进一步的约化到2^n - 1，它总是在“无穷远处”得到一个解。虽然最初的猜想是针对可行性选择和重组系统，但这里的结果推广到任何双亲传递的任意系统的可行性选择，其中包括重组和突变作为特殊情况。我们构造了一个突变选择系统的例子，它有2^n - 1个不动点，给定任何n，这表明2^n - 1是选择和传递系数的一般空间中可以找到的最尖锐的可能上限。
英文摘要：
Feldman and Karlin conjectured that the number of isolated fixed points for deterministic models of viability selection and recombination among n possible haplotypes has an upper bound of 2^n - 1. Here a proof is provided. The upper bound of 3^{n-1} obtained by Lyubich et al. (2001) using Bezout's Theorem (1779) is reduced here to 2^n through a change of representation that reduces the third-order polynomials to second order. A further reduction to 2^n - 1 is obtained using the homogeneous representation of the system, which yields always one solution `at infinity'. While the original conjecture was made for systems of viability selection and recombination, the results here generalize to viability selection with any arbitrary system of bi-parental transmission, which includes recombination and mutation as special cases. An example is constructed of a mutation-selection system that has 2^n - 1 fixed points given any n, which shows that 2^n - 1 is the sharpest possible upper bound that can be found for the general space of selection and transmission coefficients.
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