led by Josef Schicho

Symbolic computation is concerned with algorithmic manipulations of symbolic objects. These could be objects in formal language, such as formulas or programs, or algebraic objects, such as polynomials or residue classes, or geometric objects, such as lines or circles. Research in symbolic computation combines advanced mathematics with advanced computing techniques.

Linz has a strong tradition in the field of symbolic computation going back at least to the foundation of RISC by Bruno Buchberger in 1987. The symbolic computation group at RICAM works mainly on the following topics:

• Symbolic-numeric methods for polynomial system solving:
  Fo many applications of polynomial system solving - for instance in computer aided geometric design - both the given data and the requested result are modelled by floating point numbers; small numerical errors are unavoidable. On the other hand, many algorithms for polynomials, such as implicitization or parametrization, are severely ill-posed: a slight change in the coefficient may cause a qualitative change. We devise algorithms for approximate polynomials and study their numerical behavior.
  
  
• Exact methods for polynomial system solving:
  There are, on the other hand, also instances of polynomial system solving where the existence of an exact solution is guaranteed by number theory, and the size of the solutions can be controlled. Our goal is to compute the solutions exactly in these cases, and to extend the domain of problems where exact methods can be used.
  
  
• Symbolic Functional Analysis:
  Here, symbolic methods are applied to various problems in functional analysis. The main objects of study are operators between function spaces, as arising in differential equations, particularly boundary problems. Representing the relevant operator identities by noncommutative polynomials, one can apply Groebner bases for manipulating certain operators. In the case of two-point boundary problems, this leads to symbolic algorithms for solving and factoring such problems.
  
  
• Singularity analysis:
  Here, the goal is to develop tools and techniques for constructing and geometric understanding of the singular solution variety of polynomial equations. We use resolution, parametrization by multivariate power series or Puiseux series, differential forms, arc spaces.
  
  
• Piecewise algebraic representations of curves and surfaces:
  In the simulation and optimization of free form shapes, one frequently uses polynomial or rational spline functions, both implicitly, as the zero contour of spline functions, and explicitly, as image of parametrizations. We develop algorithms for the design and analysis of such shapes.
  
  

---

The Institute is named after the famous Austrian mathematician Johann Radon (1887-1956)

Medieninhaber:
Österreichische Akademie der Wissenschaften
Juristische Person öffentlichen Rechts (BGBl 569/1921 idF BGBl I 130/2003)
Dr. Ignaz Seipel-Platz 2, 1010 Wien
Diese Website dient zur Information über die wissenschaftlichen Aktivitäten der Österreichischen Akademie der Wissenschaften und setzt somit den gesetzlichen Auftrag um, die Wissenschaft in jeder Hinsicht zu fördern.

This RICAM page was made with 100% valid HTML & CSS - Send comments to Webmaster
Today's date and time is 05/22/12 - 13:55 CEST and this file ( /research/symcomp/index.html ) was last modified on 11/28/06 - 11:51 CET