> The initial point (and randomness) has more influence on the solution time than almost anything else. On that remark, a theoretical tool to choose the initia

> The initial point (and randomness) has more influence on the solution time than almost anything else. On that remark, a theoretical tool to choose the initial point is the Newton Polygon. For univariate polynomials, this leads to some of the fastest solver [1]. The idea to choose the initial point in this case is notably described in Section 6 of this paper [2]. A very rough intuition of the idea is that a polynomial p(z) doesn't vanish if one of its monomials is significantly larger than all the other. Reciprocally, the polynomial has more chances to vanish for values z such that two monomials have the same order of magnitude. Hence this is a good value to choose your initial point. Finally, deciding if two monomials have the same order of magnitude can be done by taking their log, and then it becomes geometry with polygons, hence the name Newton Polygon. [1]: https://numpi.dm.unipi.it/scientific-computing-libraries/mpsolve/ https://numpi.dm.unipi.it/scientific-computing-libraries/mps... [2]: https://woelen.homescience.net/science/math/exps/polynomials/mpsolve3_theory_paper.pdf https://woelen.homescience.net/science/math/exps/polynomials...