Forma de Chow y descomposición equidimensional

JERONIMO, GABRIELA; PUDDU, SUSANA; SABIA, JUAN

Buenos Aires

Encuentro; Primer Encuentro de Algebra del Departamento de Matemática de la FCEyN, UBA; 1999

Different problems appearing nowadays are related to systems of polynomial equations. Some of these problems can be solved simply by deciding whether the associated polynomial equation system is consistent or not. However, when the system is consistent, it is sometimes necessary to describe the set of its solutions. The set of solutions of a polynomial equation system is called an algebraic variety and has many interesting geometric properties and invariants. One of the most important invariants related to an algebraic variety is its dimension (for example, a finite set of points has dimension 0, a curve has dimension 1, a surface has dimension 2, etc.). An algebraic variety may have components of different dimensions. When all the components of an algebraic variety have the same dimension it is called equidimensional. A well-known result states that any algebraic variety V over an algebraically closed field can be decomposed into a union of equidimensional algebraic varieties V1; ... ; Vt such that dim Vi  is not equal to im Vj if i is not equal to j. Moreover, the varieties V1;... ; Vt can be chosen so that no irreducible component of Vi is contained in Vj if i is not equal to j. If this is the case, the varieties Vi are uniquely determined, each Vi  is called an equidimensional component of V and the set  {V1;...; Vt} is called the equidimensional decomposition of the algebraic variety V . There are different algorithms which describe geometric decompositions of an algebraic variety V (see [Chistov-Grigorev] for irreducible decomposition, [Giusti-Heintz 1] for both irreducible and equidimensional decomposition and [Elkadi-Mourrain] for equidimensional decomposition).and the set  {V1;...; Vt} is called the equidimensional decomposition of the algebraic variety V . There are different algorithms which describe geometric decompositions of an algebraic variety V (see [Chistov-Grigorev] for irreducible decomposition, [Giusti-Heintz 1] for both irreducible and equidimensional decomposition and [Elkadi-Mourrain] for equidimensional decomposition). In this work, we prove the existence of an algorithm that, from a finite set of polynomials defining the algebraic variety V , produces polynomials defining each equidimensional component of V . The algebraic complexity of our algorithm is lower than the complexity of any other known algorithm solving this problem.The algorithm we are going to describe here is based on computing the equidimensional decomposition of a projective variety by means of certain associated polynomials, called Chow forms (for other algorithms computing the Chow form of an equidimensional projective variety see [Giusti-Heintz 1] and [Caniglia], also see [Puddu-Sabia] in the case that the variety is irreducible). The lower complexity of this algorithm is essentially due to a special way of coding output polynomials, called straight line programs, which showed to be effective in the construction of algorithms to solve many algebraic and geometric problems.