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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 PNP[log]-completeness of deciding Zariski irreducibility, exhibits for the first time a problem with a geometric nature complete in this class.

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Correspondence to Peter Bürgisser.

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Manuscript received 3 March 2005, revised 12 March 2006

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Bürgisser, P., Cucker, F. & de Naurois, P.J. The complexity of semilinear problems in succinct representation. comput. complex. 15, 197–235 (2006). https://doi.org/10.1007/s00037-006-0213-6

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Keywords.

  • BSS additive model
  • semilinear sets
  • complete problems

Subject classification.

  • 68Q15