@Chris : sometimes we will get the career we have not expected at all ... 
-)  

Just watched last sunday one video interview with Simon (famous math, 
philantropic,  NY city market and so on .. millionnaire businessman) as 
guest : he had never predicted that his math work at young age will be 
included into the main core of modern ... physics theory.

The question "is it normal for simple non linear equations to be unsolvable 
systematically ?" is debatable. Since long time. Try to think about the 
length of arc of an ellipse. What is simple ? Sometimes we forget, the 
"closed form" of solution of equation doesn't exist, at least with current 
known formulas.

Before my post reply : I went to URL about maxima (which is running under 
the scene of sagemath) ... and of course, my POV,.., not solving 
everything...especially  some quadratic equations or in the p-adic fields 
(where is a basic Hensel lemma algorithm?)

Dominique. 

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