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.
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The authors acknowledge helpful comments of Gregory Brumfield, Jim Burke and Joe Traub for directing them to important references on the subject of algebraic plane and space curves.
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The authors dedicate this paper to Cy Derman for his enduring friendship and contributions.
Uriel G. Rothblum (1947–2012)—Deceased.
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Eaves, B.C., Rothblum, U.G. Generating approximate parametric roots of parametric polynomials. Ann Oper Res 241, 515–573 (2016). https://doi.org/10.1007/s10479-014-1534-5
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DOI: https://doi.org/10.1007/s10479-014-1534-5