Monopoles, particles and rational functions

We prove a recent conjecture of Manton and Murray: if a polynomial p(z) of degree k - 1 is given, then an SU(2) monopole corresponding to a rational function p(z)/q(z) with well-separated poles beta(1),...,beta(k) is approximately made up from charge 1 monopoles located at points (1/2 In \p(beta(i))\, beta(i)). We show how the rate of approximation changes with the numerator p(z) with the result that, as long as the values of the numerator remain close together relative to the distances between poles, the above statement remains true and ceases to be so otherwise. We also show that the spectral curve of the monopole approaches the union of curves of charge 1 monopoles exponentially fast. This remains true for SU(N) monopoles.