# Certification for Polynomial Systems via Square Subsystems

## TL;DR

This paper introduces methods for certifying solutions to overdetermined polynomial systems by leveraging square subsystems and advanced algebraic techniques, enabling reliable solution verification and completeness checks.

## Contribution

It proposes novel certification approaches for overdetermined polynomial systems using square subsystems and algebraic tools like liaison and Newton-Okounkov bodies.

## Key findings

- Effective certification of solutions to overdetermined systems.
- Methods to reject nonsolutions and verify completeness.
- Use of algebraic geometry tools for solution certification.

## Abstract

We consider numerical certification of approximate solutions to a system of polynomial equations with more equations than unknowns by first certifying solutions to a square subsystem. We give several approaches that certifiably select which are solutions to the original overdetermined system. These approaches each use different additional information for this certification, such as liaison, Newton-Okounkov bodies, or intersection theory. They may be used to certify individual solutions, reject nonsolutions, or certify that we have found all solutions.