Generating approximate parametric roots of parametric polynomials

Abstract

Built upon a ground field is the parametric field, the Puiseux field, of semi-terminating formal fractional power series. A parametric polynomial is a polynomial with coefficients in the parametric field, and roots of parametric polynomials are parametric. For a parametric polynomial with nonterminating parametric coefficients and a target accuracy, using sensitivity of the Newton Polygon process, a complete set of approximate parametric roots, each meeting target accuracy, is generated. All arguments are algebraic, from the inside out, self-contained, penetrating, and uniform in that only the Newton Polygon process is used, for both preprocessing and intraprocessing. A complexity analysis over ground field operations is developed; setting aside root generation for ground field polynomials, but bounding such, polynomial bounds are established in the degree of the parametric polynomial and the target accuracy.