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.
JDH was supported by Sloan Fellowship BR2014-110 TR14 and NSF ACI-1460032. SNS was supported by Schmitt Leadership Fellowship in Science and Engineering.
AK was partially supported by the Max Planck Institute for Mathematics in the Sciences.
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Hauenstein, J.D., Kulkarni, A., Sertöz, E.C., Sherman, S.N. (2018). Certifying Reality of Projections. In: Davenport, J., Kauers, M., Labahn, G., Urban, J. (eds) Mathematical Software – ICMS 2018. ICMS 2018. Lecture Notes in Computer Science(), vol 10931. Springer, Cham. https://doi.org/10.1007/978-3-319-96418-8_24
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DOI: https://doi.org/10.1007/978-3-319-96418-8_24
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