The complexity of semilinear problems in succinct representation

Abstract

We prove completeness results for twenty-three problems in semilinear geometry. These results involve semilinear sets given by additive circuits as input data. If arbitrary real constants are allowed in the circuit, the completeness results are for the Blum-Shub-Smale additive model of computation. If, in contrast, the circuit is constant-free, then the completeness results are for the Turing model of computation. One such result, the P NP[log] -completeness of deciding Zariski irreducibility, exhibits for the first time a problem with a geometric nature complete in this class.

Exotic Quantifiers, Complexity Classes, and Complete Problems

“…Finally, we remark that a similar classification has already been achieved in the so-called additive BSS model, without the need to introduce exotic quantifiers [12,13].…”

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

“…But clearly the corresponding lower bound does not follow from the results in [4,6,7]. Our goals in this paper are to prove that it is PSPACE-hard to decide if a complex variety is connected, and to generalise this hardness result to higher Betti numbers of fixed order.…”

“…Our proof of Theorem 1.1 uses the strategy of [4,6,7] together with some new ideas. Bürgisser and Cucker used the fact that each language in PSPACE can be decided by a symmetric Turing machine.…”

“…In [6] the sparse encoding was used, where only non-vanishing coefficients are listed together with the exponent vector of the corresponding monomial in binary. In [4,7] the semilinear sets where represented by additive circuits, which are algebraic circuits with arithmetic restricted to additions and subtractions.…”

“…In [6] the stronger result for the problem restricted to compact real algebraic sets is proved (a gap in that proof will be filled in an Appendix of this paper). In [7] also the PSPACE-completeness of the problem of deciding connectedness of the semilinear set described by an additive decision circuit is shown.…”