Certifying Zeros of Polynomial Systems Using Interval Arithmetic

We establish interval arithmetic as a practical tool for certification in numerical algebraic geometry. Our software HomotopyContinuation.jl now has a built-in function certify, which proves the correctness of an isolated nonsingular solution to a square system of polynomial equations. The implementation rests on Krawczyk’s method. We demonstrate that it dramatically outperforms earlier approaches to certification. We see this contribution as a powerful new tool in numerical algebraic geometry, which can make certification the default and not just an option.