Quantitative fundamental theorem of algebra

Abstract

Using subresultants, we modify a 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 required to hold only for real polynomials of degree at most d². 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.