So the problem has poor complexity: what next?

Some interesting problems have a poor worst-case complexity, but may still be soluble in some, or many, cases. If appropriate, there is ‘fixed-parameter tractability’ as one way of describing the solubility. We can also look at weak complexity (i.e. ignoring rare bad cases, but in a principled way). But in other cases, e.g. SAT-solving, there isn’t necessarily a neat theoretical solution, but powerful practical ideas. In particular, we look at computer algebra, and particularly real quantifier elimination.¹¹The author was partially supported by EPSRC grant EP/T015713.