As discussed in this question, I drafted a spec algorithm that hinges on finding a specific root of a system of trigonometric equations satisfying the following recurrence:
$\qquad f_{p_0} = 0\\ \qquad p_0 = 2\\ \qquad \displaystyle f_{p_n}(x) = f_{p_{n-1}}(x) + \prod_{k=2}^{p_{n-1}} (-\cos(2\pi(x+k-1)/p_{n-1}) + 1)\\ \qquad \displaystyle p_n = \min\left\{ x > p_{n-1} \mid f_{p_n}(x) = 0\right\}$
Playing with this system a bit over on Wolfram|Alpha, it seems I can get specific answers to the recurrence from their computer algebra system. Unfortunately, I can find no specific documentation on the methods they're using to solve my equations.
My question, then:
What methods (and what time and space complexities) do computer algebra systems use to solve these forms of equations? I suspect the Gröbner basis is commonly used, but I could be very wrong.
