Computing Series Expansions of Algebraic Space Curves

We work towards a series-based computational approach for polynomial systems having positive-dimensional solution sets. The tropical variety gives information on the exponents of the leading terms of the series; we provide insight into when the purely polyhedral and more easily computed tropical prevariety is sufficient. When it is not sufficient and hidden cones exist, we give an alternative to known symbolic algorithms based on polyhedral end games. We develop an effective method to apply the Gauss-Newton algorithm over power or Laurent series, using linearization and a lower triangular echelon form; we can thus extend the information obtained tropically with quadratic convergence. We also characterize when tropical methods can be avoided entirely. Finally we give applications to several problems in view of extending current approaches to homotopy continuation to allow for starting from singular solutions. We also provide a result related to the Backelin component of the cyclic-16 roots polynomial system.