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Testing Low Degree Trigonometric Polynomials

  • Martijn Baartse
  • Klaus Meer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8476)

Abstract

We design a probabilistic test verifying whether a given table of real function values corresponds to a trigonometric polynomial f : F k ↦ℝ of certain (low) degree. Here, F is a finite field. The problem is studied in the framework of real number complexity as introduced by Blum, Shub, and Smale. Our main result is at least of a twofold interest. First, it provides one of two major lacking ingredients for proving a real PCP theorem along the lines of the proof of the original PCP theorem in the Turing model. Secondly, beside the PCP framework it adds to the still small list of properties that can be tested in the BSS model over ℝ.

Keywords

Trigonometric Polynomial Univariate Restriction Algebraic Polynomial Univariate Polynomial Rejection Probability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Cambridge University Press (2009)Google Scholar
  2. 2.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and hardness of approximation problems. Journal of the ACM 45(3), 501–555 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Arora, S., Safra, S.: Probabilistic checking proofs: A new characterization of NP. Journal of the ACM 45(1), 70–122 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer (1999)Google Scholar
  5. 5.
    Baartse, M., Meer, K.: The PCP theorem for NP over the reals. To appear in Foundations of Computational Mathematics. SpringerGoogle Scholar
  6. 6.
    Baartse, M., Meer, K.: Topics in real and complex number complexity theory. In: Montana, J.L., Pardo, L.M. (eds.) Recent Advances in Real Complexity and Computation, Contemporary Mathematics, vol. 604, pp. 1–53. American Mathematical Society (2013)Google Scholar
  7. 7.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer (1998)Google Scholar
  8. 8.
    Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc. 21, 1–46 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. Springer (1997)Google Scholar
  10. 10.
    Dinur, I.: The PCP theorem by gap amplification. Journal of the ACM 54(3) (2007)Google Scholar
  11. 11.
    Friedl, K., Hátsági, Z., Shen, A.: Low-degree tests. In: Proc. SODA, pp. 57–64 (1994)Google Scholar
  12. 12.
    Meer, K.: Transparent long proofs: A first PCP theorem for NP. Foundations of Computational Mathematics 5(3), 231–255 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Meer, K.: Almost transparent short proofs for NP. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 41–52. Springer, Heidelberg (2011)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Martijn Baartse
    • 1
  • Klaus Meer
    • 1
  1. 1.Computer Science Institute, BTU Cottbus-SenftenbergCottbusGermany

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