Note on the Coincidence theorem

Abstract

We generalize Coincidence theorem due to Walsh for symmetric and linear polynomial in n complex variables, that is linear in each of them having total degre n. We discuss case when total degree is smaller then n. This case has been already studied by some authors. We obtained extension of known results that led us also to more elegant formulation of the theorem. Using that extension we discuss number of zeros of a k-th derivative of a complex polynomial, in the case when all zeros of that polynomial (except possibly one of them) are contained in a disc which center is a centroid of that zeros. In one of our previous papers we discussed case of a first derivative of a polynomial, so we now extended our results to the case of the k-th derivative.