The computational complexity of the Chow form - Universidad de ...

extended embed settings

The computational complexity of the Chow form Gabriela Jeronimo 1 , Teresa Krick 1 , Juan Sabia 1 and Martín Sombra 1,2 Abstract. We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. In particular, this gives an alternative procedure for the effective equidimensional decomposition of the variety, since each equidimensional component is characterized by its Chow form. The expected complexity of the algorithm is polynomial in the size and the geometric degree of the input equation system defining the variety. Hence it improves (or meets in some special cases) the complexity of all previous algorithms for computing Chow forms. In addition to this, we clarify the probability and uniformity aspects, which constitutes a further contribution of the paper. The algorithm is based on elimination theory techniques, in line with the geometric resolution algorithm due to M. Giusti, J. Heintz, L.M. Pardo and their collaborators. In fact, ours can be considered as an extension of their algorithm for zero-dimensional systems to the case of positive-dimensional varieties. The key element for dealing with positive-dimensional varieties is a new Poisson-type product formula. This formula allows us to compute the Chow form of an equidimensional variety from a suitable zero-dimensional fiber. As an application, we obtain an algorithm to compute a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As another application, we derive an algorithm for the computation of the (unique) solution of a generic over-determined polynomial equation system. Keywords. Chow form, equidimensional decomposition of algebraic varieties, symbolic Newton algorithm, sparse resultant, over-determined polynomial equation system. MSC 2000. Primary: 14Q15, Secondary: 68W30. Contents Introduction 2 1 Preliminaries 6 1.1 The Chow form of a quasiprojective variety . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Data and algorithm structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Complexity of basic computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Effective division procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 The representation of the Chow form 21 2.1 Newton’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2 A product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 The computation of the Chow form 31 3.1 Geometric resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Intersection of a variety with a hypersurface . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 Separation of varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4 Equations in general position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.5 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1 Partially supported by UBACyT EX-X198 and CONICET PIP 02461/01-03 (Argentina). 2 Supported by a Marie Curie Post-doctoral fellowship of the European Community Program Improving Human Research Potential and the Socio-economic Knowledge Base, contract n o HPMFCT-2000-00709. 1

4 Applications We present some algo

Corollary 4.2 Let A ⊂ N n 0 be a

Corollary 3 of [22, Chapter 3] woul

[5] L. Blum, F. Cucker, M. Shub, S.

[49] F. Macaulay, Some formulae in

Loading...

Magazine: The computational complexity of the Chow form - Universidad de ...