A polynomial homotopy is a family of polynomial systems, where the systems in the family depend on one parameter. If for one 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? Applying the ratio theorem of Fabry on the solution paths defined by the homotopy, extrapolation methods can accurately locate the nearest singularity. Once the radius of convergence is known, then via a transformation of the continuation parameter, the series expansions of the solution curves will have convergence radius equal to one. To compute all coefficients of the series we propose the quaternion Fourier transform.