  Cylindrical Algebraic Decomposition (CAD) is a key tool in computational
algebraic geometry, best known as a procedure to enable Quantifier Elimination
over real-closed fields. However, it has a worst case complexity doubly
exponential in the size of the input, which is often encountered in practice.
It has been observed that for many problems a change in algorithm settings or
problem formulation can cause huge differences in runtime costs, changing
problem instances from intractable to easy. A number of heuristics have been
developed to help with such choices, but the complicated nature of the
geometric relationships involved means these are imperfect and can sometimes
make poor choices. We investigate the use of machine learning (specifically
support vector machines) to make such choices instead.
  Machine learning is the process of fitting a computer model to a complex
function based on properties learned from measured data. In this paper we apply
it in two case studies: the first to select between heuristics for choosing a
CAD variable ordering; the second to identify when a CAD problem instance would
benefit from Groebner Basis preconditioning. These appear to be the first such
applications of machine learning to Symbolic Computation. We demonstrate in
both cases that the machine learned choice outperforms human developed
heuristics.
