By Coolidge J.

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13) is facilitated if we know part of its roots. In fact, if we know q roots a(1), ... ,a(q), then we can subtract the sum ( a j(l))P + ... + ( a j(q))P , p -_ 1, ... ,k - q, from the power sums of order p of j-th coordinates of all roots (k = k1 ... 13)). Then we obtain the power sums (of order p) of j-th coordinates of the remaining roots. 1) we can then find the polynomial whose zeros are j-th coordinates of the remaining roots. 2. Let the polynomial Q and the homogeneous polynomial P satisfy the conditions deg Q < deg P = k, and P =J 0 on a compact set Keen.

Then, by the rule for differentiation of composite functions, we have 8f3h 8gf3 (a) = 0 and for all,8, with 11,811 < m. i m l~ lit ... i m [~(f-I(O))] 8Wil .. Wi = ••• W::l •.. (211'i)n . 8mh (a). Jr(a) 81it .. 3) follows in this case. We continue the proof by induction on m. 3) has been proved for m = 1 and for a function h, such that h(a) = o. We must prove it also for a function h, which is not equal to zero at the point a. Letting h = Jr , and w = f(z), we have r r Jrdz = dw = 1rf fpf1 ... fn 1rw WpW1 ...

0 0 0 a~-1Q(a1) a~-2Q(a1) 0 Q(at} 13;-1 f3r 1 13:- 1 a n-1 a n-1 1 2 131 13. a1 a2 1 1 1 132 1 1 a~-1Q(an) a~-2Q(an) CHAPTER 1. 26 n k=l j=1 k