Certifying reality of projections
For researchers in numerical algebraic geometry, this provides a certified method to determine reality of projected solutions, addressing a gap in certification for non-square systems.
The authors extend Smale's alpha theory to certify whether projections of solutions to polynomial systems are real, and apply this to count real and totally real tritangent planes for genus-4 curves.
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.