8,237 research outputs found
An asymptotically tight bound on the number of semi-algebraically connected components of realizable sign conditions
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 polynomials in
whose degrees are at most 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.Comment: 19 pages. Bibliography has been updated and a few more references
have been added. This is the final version of this paper which will appear in
Combinatoric
- …