Multiple Roots and Approximate GCDs

The pejorative manifold was defined by Kahan in 1972. While multiple roots of a polynomial are naturally ill conditioned, we have that: if we know the multiplicities of the roots, then the roots are well conditioned. Consider a polynomial p with k distinct roots zi, the ith root has multiplicity mi, for i = 1, 2, . . . , k. The degree n of p is then the sum of multiplicities: n = m1 +m2 + · · ·+mk. We derive the relationship between the roots and coefficients of p as follows: