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An asymptotically tight bound on the number of semi-algebraically connected components of realizable sign conditions
<span title="">2009</span>
<i title="Springer Nature">
<a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ujm6quky2va7pmzbitmgsrksbu" style="color: black;">Combinatorica</a>
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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
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... 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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