Certifying reality of projections

TL;DR

This paper develops a method using Newton-invariant sets and Smale's alpha theory to certify the reality of solutions and their projections in polynomial systems, with applications to counting real tritangent planes.

Contribution

It extends certification techniques to determine the reality of projections of solutions in polynomial systems, enabling rigorous counting of real geometric configurations.

Findings

Successfully certifies the reality of projections of solutions.

Counts real and totally real tritangent planes for genus 4 curves.

Demonstrates the method's effectiveness in algebraic geometry applications.

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.