Solving systems of polynomial inequalities in subexponential time
Journal of Symbolic Computation
Journal of Symbolic Computation
Winning Concurrent Reachability Games Requires Doubly-Exponential Patience
LICS '09 Proceedings of the 2009 24th Annual IEEE Symposium on Logic In Computer Science
On the minimum of a positive polynomial over the standard simplex
Journal of Symbolic Computation
Exact algorithms for solving stochastic games: extended abstract
Proceedings of the forty-third annual ACM symposium on Theory of computing
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We prove an explicit bound on the radius of a ball centered at the origin which is guaranteed to contain all bounded connected components of a semi-algebraic set S@?R^k defined by a weak sign condition involving s polynomials in Z[X"1,...,X"k] having degrees at most d, and whose coefficients have bitsizes at most @t. Our bound is an explicit function of s,d,k and @t, and does not contain any undetermined constants. We also prove a similar bound on the radius of a ball guaranteed to intersect every connected component of S (including the unbounded components). While asymptotic bounds of the form 2^@t^d^^^O^^^(^^^k^^^) on these quantities were known before, some applications require bounds which are explicit and which hold for all values of s,d,k and @t. The bounds proved in this paper are of this nature.