VPSPACE and a Transfer Theorem over the Reals

  • Pascal Koiran
  • Sylvain Perifel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4393)


We introduce a new class VPSPACE of families of polynomials. Roughly speaking, a family of polynomials is in VPSPACE if its coefficients can be computed in polynomial space. Our main theorem is that if (uniform, constant-free) VPSPACE families can be evaluated efficiently then the class \({\sf PAR}_{\mathbb{R}}\) of decision problems that can be solved in parallel polynomial time over the real numbers collapses to P . As a result, one must first be able to show that there are \({\sf VPSPACE}\) families which are hard to evaluate in order to separate \(\sf{P}_{\mathbb{R}}\) from \(\sf{NP}_{\mathbb{R}}\), or even from \(\sf{PAR}_{\mathbb{R}}\).


computational complexity algebraic complexity  Blum-Shub-Smale model Valiant’s model 


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Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Pascal Koiran
    • 1
  • Sylvain Perifel
    • 1
  1. 1.LIP, École Normale Supérieure de Lyon 

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