Cup length as a bound on topological complexity

Polynomial solving algorithms are essential to applied mathematics and the sciences. As such, reduction of their complexity has become an incredibly important field of topological research. We present a topological approach to constructing a lower bound for the complexity of a polynomial-solving algorithm, and give a concrete algorithm to do this in the case that $\mathrm{d}\mathrm{e}\mathrm{g}(f)=2,3,4$.