Seminar Series

Ran at 10.15 on Wednesdays in 1West 2.7: Now finished.

Cylindrical Algebraic Decomposition via Triangular Decomposition

The aim of this seminar series is to understand this paper, and related topics.

Cylindrical Algebraic Decomposition via Triangular Decomposition by Chen,C., Moreno Maza,M., Xia,B. & Yang,L.. Also in Proc. ISSAC 2009.
On the Theories of Triangular Sets by Aubry,P., Lazard,D. & Moreno Maza,M. J. Symbolic Comp. 28(1999) pp. 105-124.
Triangular Sets for Solving Polynomial Systems: A Comparison of Four Methods by Aubry,P. & Moreno Maza,M. J. Symbolic Comp. 28(1999) pp. 125-154.
Comprehensive Triangular Decomposition by Chen,C., Golubitsky,O., Lemaire,F., Moreno Maza,M. & Pan,W., Proc. Computer Algebra and Symbolic Computing 2007 (eds. V.G. Ganzha, E.W. Mayr & E.V. Vorozhtsov), Springer Lecture Notes in Computer Science 4770, Springer-Verlag, 2007, pp. 73-101.
Bounds on numbers of vectors of multiplicities for polynomials which are easy to compute by Grigoriev,D.Yu. & Vorobjov,N.N.,Jr., Proc. ISSAC 2000 (ed. C. Traverso), ACM, New York, 2000, pp. 137-146. The paper referred to on 22.9.2009: Theorem 3 refers to parametric factorization of polynomials.
Ali Ayad's thesis (in French), supervised by Grigoriev.
Incomplete draft of my new book especially chapter 3.3.
Maple's 'help' on Regular Chains Really quite mathematically good, as one might expect from the ultimate authors.

18.8.2009: introduction to the problem, why non-linear is different. Concept of a triangular set. Some motivating examples.
25.8.2009: some terminology: V(T), the variety of T, W(T), the regular zeros of T. Z(P,Q), the zeros of P outside (the zeros of) Q. W(T)=Z(T,init(T)).
1.9.2009: no seminar: university closed.
8.9.2009: further discussion of the example
15.9.2009: no seminar: university Open Day
22.9.2009: initial discussion of Comprehensive Triangular Decomposition
29.9.2009: further seminar: plan for what to do in the future
Various intermediate seminars, which JHD forgot to record
28.10.2009: Marc Moreno Maza gave the seminar remotely: his slides are here.
4.11.2009: concluding seminar, led by JHD.

We have concluded with two research questions.

For the purposes of the Bradford/Davenport/Phisanbut agenda, is a full CAD necessary, or can we stop earlier in the chain indicated in Marc's slides?
This agenda (unlike many others) is agnostic with respect to variable orderings. Hence can we use the ideas of Dolzmann et al. to choose the 'best' orderings (and what exactly would 'best' be?)?