arXiv subjects

Chee K. Yap

Publications and source records attributed to Chee K. Yap.

4 recordsLinked to original sources

arXiv2021

The D-plus Discriminant and Complexity of Root Clustering

Let $p(x)$ be an integer polynomial with $m\ge 2$ distinct roots $ρ_1,\ldots,ρ_m$ whose multiplicities are $\boldsymbolμ=(μ_1,\ldots,μ_m)$. We define the D-plus discriminant of $p(x)$ to be $D^+(p):= \prod_{1\le i<j\le m}(ρ_i-ρ_j)^{μ_i+μ_j}$. We first prove a conjecture that $D^+(p)$ is a $\boldsymbolμ$-symmetric function of its roots $ρ_1,\ldots,ρ_m$. Our main result gives an explicit formula for $D^+(p)$, as a rational function of its coefficients. Our proof is ideal-theoretic, based on re-casting the classic Poisson resultant as the "symbolic Poisson formula". The D-plus discriminant first arose in the complexity analysis of a root clustering algorithm from Becker et al. (ISSAC 2016). The bit-complexity of this algorithm is proportional to a quantity $\log(|D^+(p)|^{-1})$. As an application of our main result, we give an explicit upper bound on this quantity in terms of the degree of $p$ and its leading coefficient.

cs.SC↗

arXiv2020

On mu-Symmetric Polynomials

In this paper, we study functions of the roots of a univariate polynomial in which the roots have a given multiplicity structure $μ$. Traditionally, root functions are studied via the theory of symmetric polynomials; we extend this theory to $μ$-symmetric polynomials. We were motivated by a conjecture from Becker et al.~(ISSAC 2016) about the $μ$-symmetry of a particular root function $D^+(μ)$, called D-plus. To investigate this conjecture, it was desirable to have fast algorithms for checking if a given root function is $μ$-symmetric. We designed three such algorithms: one based on Gröbner bases, another based on preprocessing and reduction, and the third based on solving linear equations. We implemented them in Maple and experiments show that the latter two algorithms are significantly faster than the first.

cs.SC↗

arXiv2017

Resolution-Exact Planner for Thick Non-Crossing 2-Link Robots

We consider the path planning problem for a 2-link robot amidst polygonal obstacles. Our robot is parametrizable by the lengths $\ell_1, \ell_2>0$ of its two links, the thickness $τ\ge 0$ of the links, and an angle $κ$ that constrains the angle between the 2 links to be strictly greater than $κ$. The case $τ>0$ and $κ\ge 0$ corresponds to "thick non-crossing" robots. This results in a novel 4DOF configuration space ${\mathbb R}^2\times ({\mathbb T}\setminusΔ(κ))$ where ${\mathbb T}$ is the torus and $Δ(κ)$ the diagonal band of width $κ$. We design a resolution-exact planner for this robot using the framework of Soft Subdivision Search (SSS). First, we provide an analysis of the space of forbidden angles, leading to a soft predicate for classifying configuration boxes. We further exploit the T/R splitting technique which was previously introduced for self-crossing thin 2-link robots. Our open-source implementation in Core Library achieves real-time performance for a suite of combinatorially non-trivial obstacle sets. Experimentally, our algorithm is significantly better than any of the state-of-art sampling algorithms we looked at, in timing and in success rate.

cs.CG↗

arXiv2015

Certified Computation of planar Morse-Smale Complexes

The Morse-Smale complex is an important tool for global topological analysis in various problems of computational geometry and topology. Algorithms for Morse-Smale complexes have been presented in case of piecewise linear manifolds. However, previous research in this field is incomplete in the case of smooth functions. In the current paper we address the following question: Given an arbitrarily complex Morse-Smale system on a planar domain, is it possible to compute its certified (topologically correct) Morse-Smale complex? Towards this, we develop an algorithm using interval arithmetic to compute certified critical points and separatrices forming the Morse-Smale complexes of smooth functions on bounded planar domain. Our algorithm can also compute geometrically close Morse-Smale complexes.

cs.CG↗