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A Polyhedral Homotopy Algorithm for Real Zeros

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Abstract

We design a homotopy continuation algorithm, that is based on Viro’s patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients that satisfy certain concavity conditions, it tracks optimal number of solution paths, and it operates entirely over the reals. In more technical terms, we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying A-discriminant amoeba. We provide a detailed exposition of connections between Viro’s patchworking method, convex geometry of A-discriminant amoeba complements, and computational real algebraic geometry.

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Notes

  1. Carried out on a MacBook Pro, Intel i5-5257U, 2.70GHz, 8GB RAM.

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Acknowledgements

We cordially thank Sascha Timme for implementing a preliminary version of the algorithm developed in this article in the software HomotopyContinuation.jl, and for his help with developing Example 4.1. We thank Matías Bender, Paul Breiding, Felipe Cucker, Mario Kummer, Gregorio Malajovich, Jeff Sommars, and Josue Tonelli-Cueto for useful discussions. The first author thanks J. Maurice Rojas for introducing him to the beautiful book [16]. First author is supported by NSF CCF 2110075, and the second author is supported by the DFG grant WO 2206/1-1.

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  1. Alperen A. Ergür, University of Texas at San Antonio, One UTSA Circle, San Antonio, TEXAS, 78249, USA

    Alperen A. Ergür

  2. Timo de Wolff, Technische Universität Braunschweig, Institut für Analysis und Algebra, AG Algebra, Universitätsplatz 2, 38106, Braunschweig, Germany

    Timo de Wolff

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  1. Alperen A. Ergür
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Ergür, A.A., Wolff, T.d. A Polyhedral Homotopy Algorithm for Real Zeros. Arnold Math J. 9, 305–338 (2023). https://doi.org/10.1007/s40598-022-00219-w

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