  We consider the extension of the method of Gauss-Newton from complex
floating-point arithmetic to the field of truncated power series with complex
floating-point coefficients. With linearization we formulate a linear system
where the coefficient matrix is a series with matrix coefficients, and provide
a characterization for when the matrix series is regular based on the algebraic
variety of an augmented system. The structure of the linear system leads to a
block triangular system. In the regular case, solving the linear system is
equivalent to solving a Hermite interpolation problem. We show that this
solution has cost cubic in the problem size. In general, at singular points, we
rely on methods of tropical algebraic geometry to compute Puiseux series. With
a few illustrative examples, we demonstrate the application to polynomial
homotopy continuation.
