The Intermediate Value Theorem for polynomials of low degree and the Fundamental Theorem of Algebra

DANIEL PERRUCCI; MARIE-FRANCOISE ROY

Rio de Janeiro

Congreso; International Congress of Mathematicians; 2018

IMU

There are many well-known proofs of the Fundamental Theorem of Algebra.Many algebraic proofs lie on the fact that the IntermediateValue Theorem holds on R, and actually, in order to prove thata polynomial of degree d in C[X] has a root in C, they apply the Intermediate Value Theorem to polynomials in R[X] of degree exponential in d. Recently, Michael Eisermann proposed a new proof for the Fundamental Theorem of Algebra, of a mixed algebraic-geometric nature, which is based onCauchy index, winding number and Sturm chains. It also lies on the the fact that the IntermediateValue Theorem holds on R.In this presentation, we show how to adapt Eisermann's proofthrough the use of subresultants polynomials, so that given any ordered field R and C = R[i],in order to prove thata polynomial of degree d in C[X] has a root in C, the Intermediate Value Theorem on R is only required to hold for polynomials in R[X] of degree bounded by d^2.