On the complexity of deciding connectedness and computing Betti numbers of a complex algebraic variety

Authors:
Peter Scheiblechner
Affiliations:
Department of Mathematics, University of Paderborn, D-33095 Paderborn, Germany
Venue:
Journal of Complexity
Year:
2007

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Abstract

We extend the lower bounds on the complexity of computing Betti numbers proved in [P. Burgisser, F. Cucker, Counting complexity classes for numeric computations II: algebraic and semialgebraic sets, J. Complexity 22 (2006) 147-191] to complex algebraic varieties. More precisely, we first prove that the problem of deciding connectedness of a complex affine or projective variety given as the zero set of integer polynomials is PSPACE-hard. Then we prove PSPACE-hardness for the more general problem of deciding whether the Betti number of fixed order of a complex affine or projective variety is at most some given integer.