Abstract This work studies the average complexity of solving structured polynomial systems that are characterised by a low evaluation cost, as opposed to dense random model previously used. Firstly, we design continuation algorithm computes, with high probability, an approximate zero system given only black-box program. Secondly, introduce universal prescribed L . Combining both, show can compu...

Polynomial systems occur in many fields of science and engineering. Polynomial homotopy continuation methods apply symbolic-numeric algorithms to solve polynomial systems. We describe the design and implementation of our web interface and reflect on the application of polynomial homotopy continuation methods to solve polynomial systems in the cloud. Via the graph isomorphism problem we organize...

How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale’s 17th problem that asked whether a polynomial bound is possible, we prove a quasi-optimal bound (input size)1`op1q. This improves upon the previously known (input size) 3 2 `op1q bound. The new algorithm relies on numerical continuation along rigid continuation path...

Discretizing systems of nonlinear algebraic differential equations yield polynomial systems. When using a fine discretization, the resulting polynomial system is often too large to solve using a direct solving approach. Our approach for solving such systems is to utilize a homotopy continuation based method arising from domain decomposition. This method solves polynomial systems arising from su...