# Software recommendation request: deciding whether a system of polynomial equations is solvable by radicals

The following system of equations comes from a very simple geometric figure I have to deal with a lot at work. Here $$r_0,r_1,r_2$$ and $$L$$ are known parameters, and the $$x_i$$s are the coordinates I'm after.

1. $$x_0^2 + x_3^2 = r_0^2$$
2. $$(x_2-L)^2 + x_4^2 = r_2^2$$
3. $$x_3 = r_1 + \tfrac{1}{2}x_5(x_0 - x_1)^2$$
4. $$x_4 = r_1 + \tfrac{1}{2}x_5(x_2 - x_1)^2$$
5. $$x_0 + x_3 x_5(x_0 - x_1) = 0$$
6. $$(x_2-L) + x_4 x_5(x_2 - x_1) = 0$$

I'm currently solving these numerically. This works perfectly well, but is (a) needlessly expensive if I don't have to do it and (b) extremely unsatisfying given the simple form of my equations.

My question is simply this: is there a piece of open-source (or otherwise free) software I can use to tell me - quickly and for certain - whether this system (and in future possibly others like it) can be solved explicitly in terms of radicals?

• No chance: see the explanation in maplesoft.com/applications/view.aspx?SID=153693 . Oct 6, 2019 at 15:52
• Thanks for this! I'll have a read and play with the demo a bit :)
– DCM
Oct 6, 2019 at 16:03
• The pdf: maplesoft.com/applications/download.aspx?SF=153693/… is really nice. I've heard of Groebner bases but I've never taken the time to learn what they are or what they're used for; nice to have an accessible intro.
– DCM
Oct 6, 2019 at 16:18
• Over the reals? Oct 7, 2019 at 7:57
• Over the reals yes.
– DCM
Oct 7, 2019 at 18:05