5,240 research outputs found
Galois groups of Schubert problems via homotopy computation
Numerical homotopy continuation of solutions to polynomial equations is the
foundation for numerical algebraic geometry, whose development has been driven
by applications of mathematics. We use numerical homotopy continuation to
investigate the problem in pure mathematics of determining Galois groups in the
Schubert calculus. For example, we show by direct computation that the Galois
group of the Schubert problem of 3-planes in C^8 meeting 15 fixed 5-planes
non-trivially is the full symmetric group S_6006.Comment: 17 pages, 4 figures. 3 references adde
Software for Exact Integration of Polynomials over Polyhedra
We are interested in the fast computation of the exact value of integrals of
polynomial functions over convex polyhedra. We present speed ups and extensions
of the algorithms presented in previous work. We present the new software
implementation and provide benchmark computations. The computation of integrals
of polynomials over polyhedral regions has many applications; here we
demonstrate our algorithmic tools solving a challenge from combinatorial voting
theory.Comment: Major updat
SMT-Solving Induction Proofs of Inequalities
This paper accompanies a new dataset of non-linear real arithmetic problems
for the SMT-LIB benchmark collection. The problems come from an automated proof
procedure of Gerhold--Kauers, which is well suited for solution by SMT. The
problems of this type have not been tackled by SMT-solvers before. We describe
the proof technique and give one new such proof to illustrate it. We then
describe the dataset and the results of benchmarking. The benchmarks on the new
dataset are quite different to the existing ones. The benchmarking also brings
forward some interesting debate on the use/inclusion of rational functions and
algebraic numbers in the SMT-LIB.Comment: Presented at the 2022 SC-Square Worksho
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