Title: Computing Tropical Prevarieties in Parallel,
Abstract: 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.