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Certificates of Positivity in the Bernstein Basis

Abstract

Let \(P\in\mathbb{Z[X]}\) be a polynomial of degree p with coefficients in the monomial basis of bit-size bounded by τ. If P is positive on [−1,1], we obtain a certificate of positivity (i.e., a description of P making obvious that it is positive) of bit-size O(p 4(τ+log 2 p)). Previous comparable results had a bit-size complexity exponential in p and τ (Powers and Reznick in Trans. Am. Math. Soc. 352(10):4677–4692, 2000; Powers and Reznick in J. Pure Appl. Algebra 164:221–229, 2001).

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Correspondence to Marie-Françoise Roy.

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Boudaoud, F., Caruso, F. & Roy, M. Certificates of Positivity in the Bernstein Basis. Discrete Comput Geom 39, 639–655 (2008). https://doi.org/10.1007/s00454-007-9042-x

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Keywords

  • Fast Algorithm
  • Discrete Comput Geom
  • Dichotomy Phase
  • General Real
  • Compression Phase