ISSAC 2006 Tutorials  
Sunday July 9, 2006

Time  Tutorial (Click on title for abstract) 

9:3012:00 (break 10:3010:45) 
CoCoA: A System for Computations in
Commutative Algebra Anna Bigatti and Lorenzo Robbiano 
13:0015:30 (break 14:0014:15) 
Triangular Decompositions of Polynomial Systems:
from Theory to Practice Marc Moreno Maza 
15:3018:00 (break 16:3016:45) 
Hybrid SymbolicNumeric Computation Erich Kaltofen and Lihong Zhi 
Schedule subject to change. 

ISSAC 2006 conference web site 
Anna Bigatti  Lorenzo Robbiano  
DIMA  Dipartimento di Matematica Universit&agr; degli Studi di Genova Via Dodecaneso, 35 16146 Genova, Italy 
DIMA  Dipartimento di Matematica Universit&agr; degli Studi di Genova Via Dodecaneso, 35 16146 Genova, Italy 

http://www.dima.unige.it/~bigatti  http://www.dima.unige.it/~robbiano 
CoCoA is a specialpurpose system for doing Computations in Commutative Algebra. It runs on all common platforms.
CoCoA's particular strengths include ideal/module operations (such as Gröbner bases, syzygies and minimal free resolutions, intersections, divisions, the radical of an ideal, etc), polynomial factorization, exact linear algebra, computing Hilbert functions, and computing with zerodimensional schemes and toric ideals.
The usefulness of these technical skills is enhanced by the mathematically natural language for describing computations. This language is readily learned by students, and enables researchers to explore and develop new algorithms without the administrative tedium necessary when using "lowlevel" languages.
Lately the CoCoA project has entered a new phase: the new design is expressly developed as a C++ library; a server and a standalone interactive system will be built on top of this library. The design should reflect the underlying mathematical structure since this will ensure that the library is natural to use.
In this tutorial we will show several applications of Computer Commutative Algebra through the use of CoCoA and CoCoALib.
Marc Moreno Maza 
ORCCA, University of Western Ontario (UWO) London, Ontario, Canada 
http://www.csd.uwo.ca/~moreno 
Triangular decompositions are one of the major tools for solving polynomial systems. For systems of algebraic equations, they provide a convenient way to describe complex solutions and a step toward isolation of real roots or decomposition into irreducible components. Combined with other techniques, they are used for these purposes by several computer algebra systems. For systems of partial differential equations, they provide the main practicable way for determining a symbolic description of the solution set. Moreover, thanks to Rosenfeld's Lemma, techniques from the algebraic case apply to the differential one [BLOP95].
Research in this area is following the natural cycle: theory, algorithms, implementation, which will be the main theme of this tutorial. We shall also concentrate on the algebraic case and mention the differential one among the applications.
The concept of a characteristic set, introduced by Ritt [Ritt32], is the cornerstone of the theory. He described an algorithm for solving polynomial systems by factoring in field extensions and computing characteristic sets of prime ideals. Wu [Wu87] obtained a factorfree adaptation of Ritt's algorithm. Several authors continued and improved Wu's approach: among them Chou, Gao [ChGa92], Gallo, Mishra [GM90] Wang [Wang93b]. Considering characteristic sets of nonprime ideals leads to difficulties that were overcome by Kalkbrener [Kalk91] and, Yang and Zhang [YaZh94] who defined particular characteristic sets, called regular chains. See also the work of Lazard and his students [ALM97]. The first part of this tutorial will be an introduction to this notion for a general audience.
Regular chains, combined with the D5 Principle [D5] and a notion of polynomial GCD [Moreno00], have also contributed to improve the efficiency of algorithms for computing triangular decompositions, as reported in [PAMMM97b]. To go further, complexity estimates of the output regular chains were needed. Such results were provided by Dahan and Schost [DaSc04]. Together with the notion of equiprojectable decomposition, they have brought the first modular algorithm for computing triangular decompositions [DMSWX05a]. The second part of this tutorial will focus on polynomial GCDs modulo regular chains. Using the RegularChains library [LeMoXi05] in Maple, we show how they are used for producing equiprojectable decompositions.
This is certainly the hot topic today. Obtaining fast algorithms for the lowlevel routines used in triangular decompositions [DMSX06] and developing implementation techniques for them [FLMS06] are the priorities that we shall discuss in the last part of this tutorial.
Erich Kaltofen  Lihong Zhi  
Dept. of Mathematics Massachusetts Institute of Technology 77 Massachusetts Ave. Cambridge, Massachusetts 021394307, USA Permanent address: Dept. of Mathematics, North Carolina State University Raleigh, North Carolina 276958205, USA 
Key Laboratory of Mathematics Mechanization Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100080, China  
http://www.kaltofen.us  http://www.mmrc.iss.ac.cn/~lzhi 
Several standard problems in symbolic computation, such as greatest common divisor and factorization of polynomials, sparse interpolation, or computing solutions to overdetermined systems of polynomial equations have nontrivial solutions only if the input coefficients satisfy certain algebraic constraints. Errors in the coefficients due to floating point roundoff or through physical measurement thus render the exact symbolic algorithms unusable. By symbolicnumeric methods one computes minimal deformations of the coefficients that yield nontrivial results. We will present hybrid algorithms and benchmark computations based on GaussNewton optimization, singular value decomposition (SVD) and structurepreserving total least squares (STLS) fitting for several of the above problems.
A significant body of results to solve those "approximate computer algebra" problems has been discovered in the past 10 years. In the Computer Algebra Handbook the section on "Hybrid Methods" concludes as follows [GKW03]:
The challenge of hybrid symbolicnumeric algorithms is to explore the effects of imprecision, discontinuity, and algorithmic complexity by applying mathematical optimization, perturbation theory, and inexact arithmetic and other tools in order to solve mathematical problems that today are not solvable by numerical or symbolic methods alone.The focus of our tutorial is on how to formulate several approximate symbolic computation problems as numerical problems in linear algebra and optimization and on software that realizes their solutions.
Our paper at this conference presents a solution to
the approximate GCD problem for several multivariate
polynomials with real or complex coefficients.
In addition, the coefficients of the minimally deformed
input coefficients can be linearly constrained.
In our tutorial we will give a precise definition
of the approximate polynomial GCD problem and we
will present techniques based on parametric optimization
(slow) and STLS or Gauss/Newton iteration (fast)
for its numerical solution.
The fast methods can
compute globally optimal solutions, but they cannot
verify global optimality. We show how to apply the constrained
approximate GCD problem to computing
the nearest singular polynomial with a root of multiplicity at
least
Our solution and implementation of the approximate factorization problem follows our approach for the approximate GCD problem. Our algorithms are based on a generalization of the differential forms introduced by W. Ruppert and S. Gao to many variables, and use SVD or STLS and Gauss/Newton optimization to numerically compute the approximate multivariate factors.
We translate a system of polynomials into a system of linear partial differential equations (PDEs) with constant coefficients. The PDEs are brought to an involutive form by symbolic prolongations and numeric projections via SVD. The solutions of the polynomial system are obtained by solving an eigenproblem constructed from the null spaces of the involutive system and its geometric projections.
This research was supported in part by the National Science Foundation of the USA under Grants CCR0305314 and CCF0514585 (Kaltofen) and OISE0456285 (Kaltofen, Yang and Zhi). This research was partially supported by NKBRPC (2004CB318000) and the Chinese National Natural Science Foundation under Grant 10401035 (Yang and Zhi).