# Ricam Wiki

### Site Tools

public:events:liwi-jan2012

# Linz-Wien workshop in February 2012

Location: St. Michael in der Obersteiermark, http://www.gasthof-eberhard.at

This workshop will follow, as an experiment, the following rules: framework_freeform_workshop.txt.

Coordinators: David Sevilla (david.sevilla@oeaw.ac.at) and Stefan Perlega.

Participants: Herwig Hauser, Eleonore Faber, Stefan Perlega, Samuel Cristóbal, Hiraku Kawanoue, Josef Schicho, David Sevilla, Gábor Hegedüs, Niels Lubbes, Angelos Mantzaflaris, Martin Weimann, Zijia Li

## Schedule

For details of the activities please see below.

 Time Activity Wednesday, 1.2.2012 Evening Arrival and dinner 21:00 - 22:00 Preliminary discussion on the Casas-Alvero Conjecture (D. Sevilla) Thursday, 2.2.2012 08:30 - 10:15 Factorisation of Motion Polynomials (J. Schicho) 10:30 - 11:45 Transversality of singular hypersurfaces (E. Faber) 12:15 - 13:15 Analysing isolated singularities via Macaulay's inverse systems (A. Mantzaflaris) 14:00 - 15:00 Lunch 15:00 - 16:30 Closed 5R linkages (G. Hegedüs, J. Schicho) Friday, 3.2.2012 08:45 - 10:15 Factorization of bivariate polynomials using singularities at infinity (M. Weimann) 10:30 - 11:45 TBD (S. Cristóbal) 12:00 - 13:00 TBD (S. Perlega) 13:00 - 14:15 Lunch 14:30 - 15:45 Exercise: discussion of abstracts (H. Hauser) 16:00 - 17:00 TBD (N. Lubbes) Evening Departure

## List of activities

Please add to this list any activity you are responsible for. Also add yourself as a participant as you please, so we can estimate (you can add or remove yourself any time!). Finally, you can ask or comment in each entry (even anonymously, if you use the guest account).

If you need LaTeX do it as follows: <latex>\frac{a}{b}</latex>. Also BibTex available! Sample: we cite MR0286317 where <bibtex> @book {MR0286317,

  AUTHOR = {Knuth, Donald E.},
TITLE = {The art of computer programming. {V}ol. 1: {F}undamental
algorithms},
SERIES = {Second printing},

            Mills, Ont},
YEAR = {1969},
PAGES = {xxi+634},
MRCLASS = {68.00 (65.00)},
MRNUMBER = {0286317 (44 \#3530)},

MRREVIEWER = {M. Muller}, } </bibtex>

#### Factorisation of Motion Polynomials

• Organized by: Josef Schicho, Gábor Hegedüs
• Type: exercise
• Duration: 75 minutes
• Minimum Requirements: basic knowledge of polynomial multiplication and factorisation, pencil & paper.

Short description: The participants are supposed to compute all factorisations of the motion polynomial

(for the LaTeX-impaired: P(t)=t^3+(-7/2i-6ej)t^2-7/2t+i-3/2ej.) Its left factors are in 1-1 correspondence with the 8 links of a certain linkage L. These links are assembled from LEGO parts. The 12 joints of L are in 1-1 correspondence with the linear factors occuring in these factorisations.

Expected Gain: the factorisation is not new, it was computed by the offerers before. But doing this exercise may result in enough insight to construct similar linkages, which may be constructed from the parts of the linkage above after disembling.

• Participants (tentative): David Sevilla, Angelos Mantzaflaris, Zijia Li

• Organized by: Gábor Hegedüs
• Type: lecture
• Duration: 60 minutes (will happen on Wednesday or Thursday)
• Minimum Requirements: basic knowledge of quaternions

Short description: We define algebraic and combinatorial bonds and as an application we classify closed 5R linkages.

• Participants (tentative): David Sevilla, Angelos Mantzaflaris, Zijia Li

#### Factorization of bivariate polynomials using singularities at infinity

• Proposed by: Martin Weimann
• Type: Talk and open problems
• Duration: 60 minutes
• Minimum Requirements: Introduction accessible to anyone. Then : toric geometry, singularities of curves, Cartier divisor, residues, logarithmic forms, cohomology.

Short description: One explains why one hopes to take advantage of the singularities at infinity of a plane curve to improve the usual algorithms for factoring bivariate polynomials. One discuss first when the singularities are solved under a toric embedding. In that case, one obtains a factorization algorithm with polynomial complexity in the volume of the Newton polytope ([wei. 2010], [wei. 2011]). But what happens if the curve remains singular at the toric infinity ? One hopes in such a case a faster algorithm, but new problems appear: problem of vanishing cohomology type and problem of complexity of resolution of singularities.

• Participants (tentative): David Sevilla, Angelos Mantzaflaris, Zijia Li, Gabor Hegedüs

#### The Casas-Alvero conjecture

• Proposed by: David Sevilla
• Type: Group work
• Duration: 2 to 3 hours, possibly split in two sessions
• Minimum Requirements: None. Familiarity with the conjecture is an asset of course.

Short description: The following is an easy fact of life: for any complex c and any natural n, the polynomial has one root in common with its derivative, one root in common with its second derivative, etc. until its (n-1)-th derivative. (Of course all of those coincide).

The converse is the Casas-Alvero conjecture: if f has degree n and Res(f,f')=0, Res(f,f'')=0 and so on until n-1, f has a root of multiplicity n.

In this activity we will explore what is known about the solution to this problem. We shall take a look at an article by Draisma and de Jong (http://www.ems-ph.org/journals/newsletter/pdf/2011-06-80.pdf) where two main ideas are explained:

• Reduction mod p, which was used in [von Bothmer et al., 2007] (article, Josef Schicho may give us a short talk about it) to produce two infinite families of degrees for which the conjecture is true. We may explore this, the claim by Draisma about two more families, possible improvements, but mainly the reason why a final proof should not be expected from these methods.
• The Gauss-Lucas theorem, an analytical tool that is very helpful; we can explore finer results that may give something new.
• For the real case, it could be useful to consider the results of Bembe and Galligo about the orders of the roots of f and its derivatives, see http://dl.acm.org/citation.cfm?doid=1993886.1993897 .

We will discuss these main ideas, work a bit on them for about half of the time, and then concentrate on solving the conjecture for the other half of the time.

#### Transversality of singular hypersurfaces

• Organized by: Eleonore Faber
• Type: lecture
• Duration: 60 minutes (will happen on Wednesday or Thursday)
• Minimum Requirements: TBD

Short description: A standard 45-minute talk with 15 minutes at the end for questions.

• Participants (tentative): M. Weimann

#### Analysing isolated singularities via Macaulay's inverse systems

• Proposed by: Angelos Mantzaflaris
• Type: Talk and discussions
• Duration: 60 minutes
• Minimum Requirements: Accessible to anyone with some background in linear & polynomial algebra.

Short description: We discuss how one can extract information on an isolated singular point of a polynomial ideal. We work on the dual polynomial ring and derive a description of the multiplicity structure in terms of local differential conditions. The algorithm is based on matrix-kernel computations, which can be carried out numerically, notably in the case of approximate inputs. One can use the local structure in applications such as: deflation of the multiplicity of the root, identification of a unique singular root in a small enough domain, computation of the topological degree and the number of half-branches attached to a real algebraic curve singularity.

• Participants (tentative): M. Weimann