Let A(x) be a polynomial A0
xn + A1 Xn-1
+ ... + An with complex coefficients and
n distinct (complex) solutions z1,...,
Also, let B(x) be a polynomial B0 xm + ... + Bm with complex coefficients. (B0 ≠ 0)
Galois theory tells us that not for all n is there a closed form function from (A0, ..., An) to (z1, ..., zn) [where by "closed form" we mean a finite composition of addition, multiplication, subtraction, division and radixes].
Part 1For which values of (n,m) is the sum of B(zi) (taken over all zi) calculable as a closed form function of (A0, ..., An, B0, ..., Bm)?
Part 2For which values of (n,m) is the product of B(zi) (taken over all zi) calculable as a closed form function of (A0, ..., An, B0, ..., Bm)?
Answer both parts to be considered a solver.
I thank Ito Kang for pointing this riddle out to me. Full credits for it will be given on the solution page.
List of solvers:Dan Dima (13 November 08:09)
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