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public:events:liwi-may2017

Linz-Wien workshop in May 2017

Location: in Klagenfurt. www.hotel-sille.com Time: From Wed(12:00), 03.05.2017 to Sat(12:00), 06.05.2017.

Unlike previous years, we are going to focus on one particular topic. The proposed topic is “Puiseux series”, and the idea is to roughly divide the meeting in two parts, a first one where the main results about Puiseux series are established, and the second, where some applications of the theory of Puiseux series are presented.

The main reference for the first part is the book “Plane Algebraic Curves” by Gerd Fisher. in particular Chapters 6 and 7.

Coordinators: Matteo Gallet (matteo.gallet@ricam.oeaw.ac.at).

Participants: Jose Capco, Giancarlo Castellano, Christopher Chiu, Matteo Gallet, Herwig Hauser, Lin Jiu, Christoph Koutschan, Hana Kovacova, Jan Legersky, Zijia Li, Niels Lubbes, Stefan Perlega, Lukas Prader, Josef Schicho.

Schedule

Date Speaker
Wednesday afternoon Niels Lubbes (Convergent) power series
Wednesday afternoon Christopher Chiu Decomposition into local branches
Thursday morning Hana Kovacova (Formal) parametrization by Puiseux series
Thursday afternoon Matteo Gallet (Convergent) parametrization by Puiseux series
Thursday afternoon Zijia Li An example of the Puiseux-Newton algorithm
Friday morning Christoph Koutschan Puiseux series and integral bases of rational functions
Friday afternoon Jan Legersky Puiseux series and Laman graphs
Friday afternoon Josef Schicho The complexity of the Newton-Puiseux algorithm

List of activities

Here is the proposed path through the theory of Puiseux series. Here we denote by [F] the book “Plane Algebraic Curves” by Fisher and by [SWP] the book “Rational Algebraic Curves: A Computer Algebra Approach” by Sendra, Winkler and Perez-Diaz.

(Convergent) power series

  • Duration: 60 minutes
  • Date: Wednesday afternoon
  • Speaker: Niels

Short description: This talk will introduce the basic concepts for power series, and will state the Weierstrass Preparation and Division Theorem. The material that could be covered is

  • [F] Section 6.1 (quickly)
  • [F] Section 6.2 (quickly)
  • [F] Section 6.3/6.4 (only the definition of convergent power series, and corollaries 1 and 2)
  • [F] Section 6.5 (only the statement of the theorem)
  • [F] Section 6.6
  • [F] Section 6.7 (the two statements of Weierstrass theorems, without proofs)
  • [F] Section 6.9 (the implicit function theorem)

Decomposition into local branches

  • Duration: 60 minutes
  • Date: Wednesday afternoon
  • Speaker: Christopher

Short description: This talk will introduce the decomposition of a germ of a curve into local branches.

  • [F] Section 6.11 (statement of the theorem and of the lemma)
  • [F] Section 6.12 (in full detail, recalling also the techniques used in Section 2.3)

(Formal) parametrization by Puiseux series

  • Duration: 90-120 minutes (this talk could be split among two people)
  • Date: Thursday morning
  • Speaker(s): Hana

Short description: This talk will prove that local branches can be parametrize by formal Puiseux series.

  • [F] Section 7.1
  • [F] Section 7.2
  • [F] Section 7.3
  • [F] Section 7.4
  • [F] Section 7.5 (without the proof of the lemma)
  • [F] Section 7.6
  • [F] Section 7.7 (without the proof of the corollary)

(Convergent) parametrization by Puiseux series

  • Duration: 90 minutes
  • Date: Thursday afternoon
  • Speaker(s): Matteo

Short description: This talk will prove that local branches can be parametrize by convergent Puiseux series.

  • [F] Section 7.8
  • [F] Section 7.9, all four lemmas (recalling Hensel's lemma from Section 6.10), maybe skipping the proof of Lemma 4
  • [F] Section 7.10
  • [F] Section 7.11

An example of the Puiseux-Newton algorithm

  • Duration: 45 minutes
  • Date: Thursday afternoon / Friday morning
  • Speaker: Zijia

Short description: This talk will show an example of the Puiseux-Newton algorithm.

  • [SWP] Section 2.5.2.

Puiseux series and integral bases of rational functions

  • Duration: 45 minutes
  • Date: Friday morning
  • Speaker: Christoph

Puiseux series and Laman graphs

  • Duration: 45 minutes
  • Date: Friday afternoon
  • Speaker: Jan

The complexity of the Newton-Puiseux algorithm

  • Duration: 45 minutes
  • Date: Friday afternoon
  • Speaker: Josef

Short description: Two questions arise and will be treated in this talk:

  • How should you describe the complexity of an algorithm computing infinite series? Here, we say such an algorithm has polynomial complexity if, for each n, the time/space requested to compute the nth term is polynomial.
  • Once it is established that the number of arithmetic operations is polynomial, can we also control the bit complexity? We will try to answer this question without nit-picking.
/var/www/wiki/htdocs/data/pages/public/events/liwi-may2017.txt · Last modified: 2017/04/27 10:08 by Zijia Li