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Summary: Proof of the Feldman-Karlin Conjecture on the
Maximum Number of Equilibria in an
Evolutionary System
Lee Altenberg
University of Hawai`i at Manoa
altenber@hawaii.edu
Abstract
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 2n
- 1. Here a proof is provided. The upper
bound of 3n-1
obtained by Lyubich et al. (2001) using Bézout's Theorem (1779) is reduced here to 2n
through a change
of representation that reduces the third-order polynomials to second order. A further reduction to 2n
- 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 2n
- 1 fixed points given any n, which shows that 2n
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