Exact and certified numeric computations produce rigorous and reliable results which are vital in algebraic geometry, geometric computing, cryptography, program verification, and many other areas of science and engineering. This session aims at bringing researchers from Mathematics and Computer Science together with software developers and engineers to present advances and on-going work on the development of software for symbolic and certified numeric computations with nonlinear, algebraic, differential, geometric, and hybrid systems.
A short abstract will appear on the permanent conference web page (see below) as soon as accepted.
An extended abstract will appear on the permanent conference web page (see below) as soon as accepted.
It will also appear on the proceedings that will be distributed during the meeting.
If you would like to give a talk at ICMS, you need to submit first a short abstract and then later an extended abstract. See the guideline for the details.
Abstract: We present an implementation of arbitrary-precision numerical integration with rigorous error bounds in the Arb library. Rapid convergence is ensured for piecewise complex analytic integrals by use of the Petras algorithm, which combines adaptive bisection with adaptive Gaussian quadrature where error bounds are determined via complex magnitudes without evaluating derivatives. The code is general, easy to use, and efficient, often outperforming existing non-rigorous software.
The tropical variety of a polynomial ideal over a field with
non-trivial valuation is the image of its zeroset under componentwise
valuation. It is the support of a polyhedral complex and commonly
described as the combinatorial shadow of its algebraic counterpart. A
tropical basis is a generating set of the ideal, whose intersected
tropical hypersurfaces yield the tropical variety.
In this talk, we will discuss some techniques that verify whether a given generating set is a tropical basis for zero-dimensional ideal using triangular decomposition, and how they can be used for positive-dimensional ideals. Moreover, we discuss some applications to del-Pezzo surfaces, Delta-matroids and homotopy continuation for solving systems of polynomial equations.
This is joint work with Paul Goerlach and Jeff Sommars.
Abstract: We develop algorithms to certify solutions to systems of equations involving D-finite functions. (Joint with Michael Burr and Kisun Lee)
Abstract: We describe Ccluster, a software for computing natural ε-clusters of complex roots in a given box of the complex plane. This algorithm from Becker et al. (2017) is near-optimal when applied to the benchmark problem of isolating all complex roots of an integer polynomial. It is one of the first implementations of a near-optimal algorithm for complex roots. We describe some low level techniques for speeding up the algorithm. Its performance is compared with the well-known MPSolve library and Maple.
Abstract: For two polynomials F(u, x) and G(u, x) in the variable x and the parameters u with rational coefficients, the parametric GCD (greatest common divisor) of F and G with respect to x is defined by a partition of the space of parameters u, possibly with constraints, such that for each subspace of the partition there exists a polynomial H(u,x) such that the GCD of F(a,x) and G(a,x) is equal to H(a,x) for all parametric values a of u in the subspace. Parametric GCD is a nontrivial generalization of the GCD of multivariate polynomials and the regular GCD in triangular decomposition. A simple method is proposed for computing the parametric GCD of any two polynomials by using subresultant regular subchains. The method has been implemented in Maple and included in the software package Epsilon for triangular decomposition. Experimental results are reported to demonstrate the effectiveness of the method with our implementation.