On the intrinsic complexity of elimination problems in effective Algebraic Geometry 1 Joos Heintz 2 , Bart Kuijpers 3 , Andr´ es Rojas Paredes 4 Abstract The representation of polynomials by arithmetic circuits evaluating them is an alternative data structure which allowed considerable progress in poly- nomial equation solving in the last fifteen years. We present a circuit based computation model which captures all known symbolic elimination algorithms in effective algebraic geometry and show the intrinsically exponential complex- ity character of elimination in this complexity model. 1 Introduction Modern elimination theory starts with Kronecker’s 1882 paper [Kro82] where the argumentation is essentially constructive, i.e., algorithmic. Questions of efficiency of algorithms become only indirectly and marginally addressed in this paper. However, later criticism of Kronecker’s approach to algebraic geometry emphasized the algo- rithmic inefficiency of his argumentation ([Mac16], [vdW50]). In a series of more recent contributions, that started with [CGH89] and ended up with [GHM + 98], [GHH + 97], [HMW01] and [GLS01], it became apparent that this criticism is based on a too narrow interpretation of Kronecker’s elimination method. In fact, these contributions are, implicitly or explicitly, based on this method, nonwithstanding that they also contain other views and ideas coming from commutative algebra and algebraic complexity theory. A turning point was achieved by the combination of a new, global view of New- ton iteration with Kronecker’s method ([GHM + 98], [GHH + 97]). The outcome was that elimination polynomials, although hard to represent by their coefficients, allow a reasonably efficient encoding by evaluation algorithms. This circumstance sug- gests to represent in elimination algorithms polynomials not by their coefficients but by arithmetic circuits (see [HS81], [Kal88] and [FIK86] for the first steps in 1 Research partially supported by the following Argentinian, Belgian and Spanish grants: CON- ICET PIP 2461/01, UBACYT 20020100100945, PICT–2010–0525, FWO G.0344.05, MTM2010- 16051. 2 Departamento de Computaci´ on, Universidad de Buenos Aires and CONICET, Ciudad Univer- sitaria, Pab. I, 1428 Buenos Aires, Argentina, and Departamento de Matem´ aticas, Estad´ ıstica y Computaci´ on, Facultad de Ciencias, Universidad de Cantabria, Avda. de los Castros s/n, 39005 Santander, Spain. joos@dc.uba.ar & joos.heintz@unican.es 3 Database and Theoretical Computer Science Research Group, Hasselt University, Agoralaan, Gebouw D, 3590 Diepenbeek, Belgium. bart.kuijpers@uhasselt.be 4 Departamento de Computaci´ on, Universidad de Buenos Aires, Ciudad Universitaria, Pab. I, 1428 Buenos Aires, Argentina. arojas@dc.uba.ar 1 arXiv:1201.4344v1 [cs.CC] 20 Jan 2012