Certifying zeros of polynomial systems using interval arithmetic

Abstract

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 powerful new tool in numerical algebraic geometry, that can make certification the default and not just an option.

Figures (2)

• Figure 1: The picture shows two straight-line programs for evaluating the polynomial $f(x,y,z)=(x+y)z$. Let $I=([-1, 0],\,[1, 1],\, [0, 1])^T$. Then, the program on the left evaluated at $I$ yields $f(I) = ( [-1, 0] + [1, 1] ) [0, 1] = [0,1]$, while the program on the right yields $f(I) = [-1, 0] [0, 1]+ [1, 1][0, 1] = [-1,1]$.
• Figure 2: Screenshot from a Julia session, where we certify the 3264 real conics for the totally real arrangement from 3264. Here, F is the system of polynomials from (12) in 3264. The screenshot also demonstrates the simple syntax of our implementation.

Theorems & Definitions (20)

• Theorem 2.1
• Definition 3.1: Interval enclosure
• Definition 3.2: Interval matrix norm
• Definition 3.3
• Remark 3.4
• Remark 3.5
• Theorem 3.6
• Definition 3.7
• Remark 3.8
• Definition 3.9