A solution to certain polynomial equations with applications to nonlinear fitting

We present a combinatorial method for solving a certain system of polynomial equations of Vandermonde type in 2N variables by reducing it to the problem of solving two special linear systems of size N and rooting a single univariate polynomial of degree N . Over C, all solutions can be found with fixed precision using, up to polylogarithmic factors, O(N2) bitwise operations in the worst case. Furthermore, if the data is well conditioned, then this can be reduced to O(N) bit operations, up to polylogarithmic factors. As an application, we show how this can be used to fit data to a complex exponential sum with N terms in the same, nearly optimal, time.