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
This improves the best upper bound known previously which was
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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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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DOI: https://doi.org/10.1007/s00493-009-2357-x