## November 2009 riddle

Let A(x) be a polynomial A0 xn + A1 Xn-1 + ... + An with complex coefficients and n distinct (complex) solutions z1,..., zn.

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].

The riddle:

#### Part 1

For 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 2

For 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)

Elegant and original solutions can be submitted to the puzzlemaster at riddles brand.scso.com. Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.

The solution will be published at the end of the month.

Enjoy!