Symmetric subresultants and applications

Schur’s transforms of a polynomial are used to count its roots in the unit disk. These are generalized then by introducing the sequence of symmetric subresultants of two polynomials. Although they do have a determinantal definition, we show that they satisfy a structure theorem which allows us to compute them with a type of Euclidean division. As a consequence, a fast algorithm based on a dichotomic process and FFT is designed. We prove also that these symmetric subresultants have a deep link with Toeplitz matrices. Finally, we propose a new algorithm of inversion for such matrices. It has the same cost as those already known; however it is fraction free and consequently well adapted to computer algebra.