 
Summary: Proof of the FeldmanKarlin 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 3n1
obtained by Lyubich et al. (2001) using Bézout's Theorem (1779) is reduced here to 2n
through a change
of representation that reduces the thirdorder 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 biparental transmission, which includes recombination and mutation as special cases. An
example is constructed of a mutationselection system that has 2n
 1 fixed points given any n, which shows that 2n
