  A polynomial homotopy is a family of polynomial systems, where the systems in
the family depend on one parameter. If for one value of the parameter we know a
regular solution, then what is the nearest value of the parameter for which the
solution in the polynomial homotopy is singular? For this problem we apply the
ratio theorem of Fabry. Richardson extrapolation is effective to accelerate the
convergence of the ratios of the coefficients of the series expansions of the
solution paths defined by the homotopy. For numerical stability, we recondition
the homotopy. To compute the coefficients of the series we propose the
quaternion Fourier transform. We locate the closest singularity computing at a
regular solution, avoiding numerical difficulties near a singularity.
