Quantitative Fundamental Theorem of Algebra

TL;DR

This paper provides a quantitative analysis of the Fundamental Theorem of Algebra using subresultants, showing the degree bounds of the Intermediate Value Theorem needed for different proofs and highlighting their fundamental differences.

Contribution

It introduces a modified proof using subresultants that quantifies the degree of polynomials for which the Intermediate Value Theorem is required in the proof of the FTA.

Findings

FTA proof requires IVT for polynomials of degree up to d^2

Laplace's proof requires IVT for exponentially high degree

Quantitative bounds reveal differences in proof methods

Abstract

Using subresultants, we modify a recent real-algebraic proof due to Eisermann of the Fundamental Theorem of Algebra ([FTA]) to obtain the following quantitative information: in order to prove the [FTA] for polynomials of degree $d$, the Intermediate Value Theorem ([IVT]) is requested to hold for real polynomials of degree at most $d^2$. We also explain that the classical proof due to Laplace requires [IVT] for real polynomials of exponential degree. These quantitative results highlight the difference in nature of these two proofs.