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Numerically Computing Real Points on Algebraic Sets

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Abstract

Given a polynomial system f, a fundamental question is to determine if f has real roots. Many algorithms involving the use of infinitesimal deformations have been proposed to answer this question. In this article, we transform an approach of Rouillier, Roy, and Safey El Din, which is based on a classical optimization approach of Seidenberg, to develop a homotopy based approach for computing at least one point on each connected component of a real algebraic set. Examples are presented demonstrating the effectiveness of this parallelizable homotopy based approach.

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Acknowledgements

The author would like to thank Mohab Safey El Din, Charles Wampler, and the anonymous referee for their helpful comments as well as the Institut Mittag-Leffler (Djursholm, Sweden) for support and hospitality when working on this article.

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  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Jonathan D. Hauenstein

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  1. Jonathan D. Hauenstein
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Correspondence to Jonathan D. Hauenstein.

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This author was partially supported by Institut Mittag-Leffler (Djursholm, Sweden), NSF grants DMS-0915211 and DMS-1114336, and DOE ASCR grant DE-SC0002505.

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Hauenstein, J.D. Numerically Computing Real Points on Algebraic Sets. Acta Appl Math 125, 105–119 (2013). https://doi.org/10.1007/s10440-012-9782-3

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