Cup length as a bound on topological complexity

TL;DR

This paper introduces a topological method to establish lower bounds on the complexity of polynomial-solving algorithms, providing concrete algorithms for degrees 2, 3, and 4.

Contribution

It presents a novel topological approach to bound the complexity of polynomial algorithms and offers explicit algorithms for degrees 2, 3, and 4.

Findings

Established a topological lower bound for polynomial-solving complexity

Developed algorithms for degrees 2, 3, and 4

Enhanced understanding of polynomial algorithm complexity

Abstract

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 $\mathrm{deg}(f) = 2,3,4$.