The Number of Distinct Roots of a Polynomial and Its Derivatives
نویسنده
چکیده
THEOREM 1. Let f(z) be a polynomial of degree N with complex coefficients, and let / (r) (z) denote the rth derivative of f(z). If f{z) has two or more distinct roots, N Yl / (r) (z) has at least N+l distinct roots.r=0 plicity of z s as a root of/ (r) (z), for r = 0,. be the sum of the mth powers of the roots of/ (r) (z), and let A m (r) be the elementary symmetric polynomial of degree m in the roots of/ (r) (z). For m > N — r, we define A m (r) = 0. Suppose N m= 0 then / • rrw N ^ r , xm (N-m)\ N _ m _ r J w m = o v ; m (N-m-r)\ and so This defines A m (r) for all r as a polynomial in r of degree w. Note that A m (N) = 0. Newton's formula J 0 z c r m f c (K k = 1 is valid for r = 1, ...,N— 1. Solving it, we can obtain an expression for <r m (r) as a sum of terms like A kl (r) ...
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