Rigid continuation paths II. structured polynomial systems

Abstract This work studies the average complexity of solving structured polynomial systems that are characterised by a low evaluation cost, as opposed to dense random model previously used. Firstly, we design continuation algorithm computes, with high probability, an approximate zero system given only black-box program. Secondly, introduce universal prescribed L . Combining both, show can compute n equations degree at most ${D}$ in variables $\operatorname {poly}(n, {D}) L$ operations probability. exceeds expectations implicit Smale’s 17th problem.