We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy based solvers in terms of decorated graphs. Under the theoretical assumption that monodromy actions are generated uniformly, we show that the expected number of homotopy paths tracked by an algorithm following this framework is linear in the number of solutions. We demonstrate that our software implementation is competitive with the existing state-of-the-art methods implemented in other software packages.
We introduce the notion of tropical defects, certificates that a system of polynomial equations is not a tropical basis, and provide algorithms for finding them around affine spaces of complementary dimension to the zero set. We use these techniques to solve open problems regarding del Pezzo surfaces of degree 3 and realizability of valuated gaussoids of rank 4.
Pretropisms are candidates for the leading exponents of Puiseux series that represent positive dimensional solution sets of polynomial systems. We propose a new algorithm to both horizontally and vertically prune the tree of edges of a tuple of Newton polytopes. We provide experimental results with our preliminary implementation in Sage that demonstrates that our algorithm compares favorably to the definitional algorithm.
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 and classify the polynomial systems we solved. The classification with the canonical form of a graph identifies newly submitted systems with systems that have already been solved.
The computation of the tropical prevariety is the first step in the application of polyhedral methods to compute positive dimensional solution sets of polynomial systems. In particular, pretropisms are candidate leading exponents for the power series developments of the solutions. The computation of the power series may start as soon as one pretropism is available, so our parallel computation of the tropical prevariety has an application in a pipelined solver.We present a parallel implementation of dynamic enumeration. Our first distributed memory implementation with forked processes achieved good speedups, but quite often resulted in large variations in the execution times of the processes. The shared memory multithreaded version applies work stealing to reduce the variability of the run time. Our implementation applies the thread safe Parma Polyhedral Library (PPL), in exact arithmetic with the GNU Multiprecision Arithmetic Library (GMP), aided by the fast memory allocations of TCMalloc.Our parallel implementation is capable of computing the tropical prevariety of the cyclic 16-roots problem. We also report on computational experiments on the n-body and n-vortex problems; our computational results compare favorably with Gfan.
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