Service interruption on Monday 11 July from 12:30 to 13:00: all the sites of the CCSD (HAL, EpiSciences, SciencesConf, AureHAL) will be inaccessible (network hardware connection).
Skip to Main content Skip to Navigation
Preprints, Working Papers, ...

Interpolation in Valiant's theory

Abstract : We investigate the following question: if a polynomial can be evaluated at rational points by a polynomial-time boolean algorithm, does it have a polynomial-size arithmetic circuit? We argue that this question is certainly difficult. Answering it negatively would indeed imply that the constant-free versions of the algebraic complexity classes VP and VNP defined by Valiant are different. Answering this question positively would imply a transfer theorem from boolean to algebraic complexity. Our proof method relies on Lagrange interpolation and on recent results connecting the (boolean) counting hierarchy to algebraic complexity classes. As a byproduct we obtain two additional results: (i) The constant-free, degree-unbounded version of Valiant's hypothesis that VP and VNP differ implies the degree-bounded version. This result was previously known to hold for fields of positive characteristic only. (ii) If exponential sums of easy to compute polynomials can be computed efficiently, then the same is true of exponential products. We point out an application of this result to the P=NP problem in the Blum-Shub-Smale model of computation over the field of complex numbers.
Document type :
Preprints, Working Papers, ...
Complete list of metadata

Cited literature [21 references]  Display  Hide  Download
Contributor : Sylvain Perifel Connect in order to contact the contributor
Submitted on : Monday, October 1, 2007 - 3:27:15 PM
Last modification on : Saturday, September 11, 2021 - 3:17:34 AM
Long-term archiving on: : Friday, April 9, 2010 - 5:02:50 PM


Files produced by the author(s)


  • HAL Id : ensl-00175862, version 1
  • ARXIV : 0710.0360



Pascal Koiran, Sylvain Perifel. Interpolation in Valiant's theory. 2007. ⟨ensl-00175862⟩



Record views


Files downloads