Certifying reality of projections

Jonathan Hauenstein, Avinash Kulkarni, Emre Sertöz, and Samantha Sherman

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Submission date: 13. Apr. 2018
published in: Mathematical software ICMS 2018 : 6th international conference, South Bend, IN, USA, July 24-27, 2018, proceedings / J. Davenport ... (eds.)
Cham : Springer, 2018. - P. 200 - 208
(Lecture notes in computer science ; 10931) 
DOI number (of the published article): 10.1007/978-3-319-96418-8_24
Bibtex
MSC-Numbers: 14Q99, 14P05, 14Q05
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
Computational tools in numerical algebraic geometry can be used to numerically approximate solutions to a system of polynomial equations. If the system is well-constrained (i.e., square), Newton's method is locally quadratically convergent near each nonsingular solution. In such cases, Smale's alpha theory can be used to certify that a given point is in the quadratic convergence basin of some solution. This was extended to certifiably determine the reality of the corresponding solution when the polynomial system is real. Using the theory of Newton-invariant sets, we certifiably decide the reality of projections of solutions. We apply this method to certifiably count the number of real and totally real tritangent planes for instances of curves of genus 4.