Section: Scientific Foundations
Gröbner basis and triangular sets
Participants : J.C. Faugère, G. Renault, M. Safey El Din, P.J. Spaenlehauer, D. Wang, C. Mou, J. Svartz.
Let us denote by
The ideal
One Gröbner basis' main property is to provide an algorithmic method for deciding if a polynomial belongs or not to an ideal through a reduction function denoted "
If
- (i)
a polynomial
belongs to if and only if , - (ii)
Reduce(
, , ) does not depend on the order of the polynomials in the list , thus, this is a canonical reduced expression modulus , and the Reduce function can be used as a simplification function.
Gröbner bases are computable objects. The most popular method for computing them is Buchberger's algorithm ( [47] , [46] ). It has several variants and it is implemented in most of general computer algebra systems like Maple or Mathematica. The computation of Gröbner bases using Buchberger's original strategies has to face to two kind of problems :
-
(A) arbitrary choices : the order in which are done the computations has a dramatic influence on the computation time;
-
(B) useless computations : the original algorithm spends most of its time in computing 0.
For problem (A), J.C. Faugère proposed ([4] - algorithm
For problem (B), J.C. Faugère proposed ([3] ) a new criterion for detecting useless computations. Under some regularity conditions on the system, it is now proved that the algorithm do never perform useless computations.
A new algorithm named
We pay a particular attention to Gröbner bases computed for elimination orderings since they provide a way of "simplifying" the system (equivalent system with a structured shape). A well known property is that the zeros of the first non null polynomial define the Zariski closure (classical closure in the case of complex coefficients) of the projection on the coordinate's space associated with the smallest variables.
Such kinds of systems are algorithmically easy to use, for computing numerical approximations of the solutions in the zero-dimensional case or for the study of the singularities of the associated variety (triangular minors in the Jacobian matrices).
Triangular sets have a simplier structure, but, except if they are linear, algebraic systems cannot, in general, be rewritten as a single triangular set, one speaks then of decomposition of the systems in several triangular sets.
Lexicographic Gröbner bases | Triangular sets |
Triangular sets appear under various names in the field of algebraic systems. J.F. Ritt ( [64] ) introduced them as characteristic sets for prime ideals in differential algebra. His constructive algebraic tools were adapted by W.T. Wu in the late seventies for geometric applications. The concept of regular chain (see [56] and [74] ) is adapted for recursive computations in a univariate way.
It provides a membership test and a zero-divisor test for the strongly
unmixed dimensional ideal it defines. Kalkbrenner defined regular
triangular sets and showed how to decompose algebraic varieties as a
union of Zariski closures of zeros of regular triangular sets. Gallo
showed that the principal component of a triangular decomposition can
be computed in
D. Lazard contributed to the homogenization of the work completed in this field by proposing a series of specifications and definitions gathering the whole of former work [1] . Two essential concepts for the use of these sets (regularity, separability) at the same time allow from now on to establish a simple link with the studied varieties and to specify the computed objects precisely.
A remarkable and fundamental property in the use we have of the triangular sets is that the ideals induced by regular and separable triangular sets, are radical and equidimensional. These properties are essential for some of our algorithms. For example, having radical and equidimensional ideals allows us to compute straightforwardly the singular locus of a variety by canceling minors of good dimension in the Jacobian matrix of the system. This is naturally a basic tool for some algorithms in real algebraic geometry [2] , [7] , [67] .
In 1993, Wang [70] proposed a method for decomposing any polynomial system into fine triangular systems which have additional properties such as the projection property that may be used for solving parametric systems (see Section 3.4.2 ).
Triangular sets based techniques are efficient for specific problems, but the implementations of direct decompositions into triangular sets do not currently reach the level of efficiency of Gröbner bases in terms of computable classes of examples. Anyway, our team benefits from the progress carried out in this last field since we currently perform decompositions into regular and separable triangular sets through lexicographical Gröbner bases computations.