Rigid homotopies for sampling from algebraic varieties: a Waring structure complexity model
This work advances the theoretical understanding of homotopy continuation methods for polynomial system solving, a core problem in algebraic geometry and computational mathematics.
The authors prove a new complexity bound for rigid homotopies applied to polynomial systems with Waring representations of prescribed length, and present the first computational experiments with a preliminary implementation.
Polynomial system solving has seen major progress in both theory and practice over the past decade. A landmark achievement was addressing Smale's 17th problem, establishing average-case polynomial-time algorithms for computing approximate solutions of polynomial systems via homotopy continuation. Recent improvements in complexity bounds for these algorithms led to the development of rigid homotopy methods. In this article, we prove a new complexity result for rigid homotopies for polynomial systems with Waring representations of prescribed length. In addition, we provide the first computational experiments for rigid homotopies using a preliminary implementation.