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 onlyfor 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 thedifference in nature of these two proofs.