Data Structures and Polynomial Equation Solving ⋆ D. Castro 1 , M. Giusti 2 , J. Heintz 134 , G. Matera 56 , and L.M. Pardo 12 1 Depto. de Matem´aticas, Estad´ ıstica y Computaci´on, Facultad de Ciencias, Universidad de Cantabria, E-39071 Santander, Spain 2 Laboratoire GAGE, Ecole Polytechnique, F-91128 Palaiseau Cedex, France 3 Depto. de Matem´aticas, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabell´on I (1428) Buenos Aires, Argentina 4 Member of the National Council of Science and Technology (CONICET), Argentina 5 Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento, Roca 850 (1663), San Miguel, Argentina 6 Departamento de Computaci´on, Universidad Favaloro, Belgrano 1723 (1093) Buenos Aires, Argentina Abstract. Elimination theory is at the origin of algebraic geometry in the 19-th century and deals with algorithmic solving of multivariate poly- nomial equation systems over the complex numbers, or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic sequential time complexity of universal elimination proce- dures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e. polynomial equation systems) and ad- mitting the representation of certain limit objects. Our main result is the following: let be given such a data structure and together with this data structure a universal elimination algorithm, say P , solving arbitrary parametric polynomial equation systems. Suppose that the algorithm P avoids ”unnecessary” branchings and that P admits the efficient computation of certain natural limit objects (as e.g. the Zariski closure of a given constructible algebraic set or the parametric greatest common divisor of two given algebraic families of univariate polynomials). Then P cannot be a polynomial time algorithm. The paper contains different variants of this result which are formulated and discussed both from the point of view of exact (i.e. symbolic) as well as from the point of view of approximative (i.e. numeric) computing. The mentioned results shall only be discussed informally. Proofs will appear elsewhere. Keywords. Polynomial equation solving, elimination theory, complexity, continuous data structure. ⋆ Research was partially supported by the following Argentinian, European, French and Spanish grants : UBACyT UBA X–198, UNLP X–272, PIP CONICET 4571, BMBF–SETCIP AL/A97-EXI/09, ECOS A99E06, CNRS 1026 MEDICIS, DGCYT BFM2000–0349, Cooperaci´on Cient´ ıfica con Iberoamerica and HF-1999-055