An asymptotically tight bound on the number of semi-algebraically connected components of realizable sign conditions

Abstract

We prove an asymptotically tight bound (asymptotic with respect to the number of polynomials for fixed degrees and number of variables) on the number of semi-algebraically connected components of the realizations of all realizable sign conditions of a family of real polynomials. More precisely, we prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of a family of s polynomials in R[X 1, …,X k ] whose degrees are at most d is bounded by

$$ \frac{{(2d)^k }} {{k!}}s^k + O(s^{k - 1} ). $$

This improves the best upper bound known previously which was

$$ \frac{1} {2}\frac{{(8d)^k }} {{k!}}s^k + O(s^{k - 1} ). $$

The new bound matches asymptotically the lower bound obtained for families of polynomials each of which is a product of generic polynomials of degree one.

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Correspondence to Saugata Basu.

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Supported in part by an NSF Career Award 0133597 and an Alfred P. Sloan Foundation Fellowship.

Supported in part by NSA grant MDA904-01-1-0057 and NSF grants CCR-9732101 and CCR-0098246.

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Basu, S., Pollack, R. & Roy, MF. An asymptotically tight bound on the number of semi-algebraically connected components of realizable sign conditions. Combinatorica 29, 523–546 (2009). https://doi.org/10.1007/s00493-009-2357-x

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Mathematics Subject Classification (2000)

  • 14P10
  • 14P25
  • 46E25
  • 20C20