Acceleration of Subdivision Root-Finding for Sparse Polynomials

Univariate polynomial root-finding has been studied for four millennia and is still the subject of intensive research. Hundreds of efficient algorithms for this task have been proposed. A recent front-running algorithm relies on subdivision iterations. Already its initial implementation of 2018 has competed for user’s choice for root-finding in a region that contains a small number of roots. Recently, we significantly accelerated the basic blocks of these iterations, namely root-counting and exclusion tests. In [18], we solidified this approach and made our acceleration dramatic in the case of sparse polynomials and other ones defined by a black box for their fast evaluation. Our techniques are novel and should be of independent interest. In the present paper and its companion [19], we expose a substantial part of that work.