The Fundamental Theorem of Algebra in ACL2

TL;DR

This paper formalizes the proof of the Fundamental Theorem of Algebra within ACL2(r), demonstrating that every non-constant polynomial has a root by verifying properties of complex functions, continuity, and polynomial bounds.

Contribution

It introduces a novel approach using a 'context' argument in ACL2(r) to simplify proofs of classical properties involving unbound parameters.

Findings

Complex functions achieve minima over closed regions.

Complex polynomials are continuous and attain their minimum.

Non-constant polynomials have bounds on their norm away from the origin.

Abstract

We report on a verification of the Fundamental Theorem of Algebra in ACL2(r). The proof consists of four parts. First, continuity for both complex-valued and real-valued functions of complex numbers is defined, and it is shown that continuous functions from the complex to the real numbers achieve a minimum value over a closed square region. An important case of continuous real-valued, complex functions results from taking the traditional complex norm of a continuous complex function. We think of these continuous functions as having only one (complex) argument, but in ACL2(r) they appear as functions of two arguments. The extra argument is a "context", which is uninterpreted. For example, it could be other arguments that are held fixed, as in an exponential function which has a base and an exponent, either of which could be held fixed. Second, it is shown that complex polynomials are continuous, so the norm of a complex polynomial is a continuous real-valued function and it achieves its minimum over an arbitrary square region centered at the origin. This part of the proof benefits from the introduction of the "context" argument, and it illustrates an innovation that simplifies the proofs of classical properties with unbound parameters. Third, we derive lower and upper bounds on the norm of non-constant polynomials for inputs that are sufficiently far away from the origin. This means that a sufficiently large square can be found to guarantee that it contains the global minimum of the norm of the polynomial. Fourth, it is shown that if a given number is not a root of a non-constant polynomial, then it cannot be the global minimum. Finally, these results are combined to show that the global minimum must be a root of the polynomial. This result is part of a larger effort in the formalization of complex polynomials in ACL2(r).