### Bounding the radii of balls meeting every connected component of semi-algebraic sets [article]

Saugata Basu, Marie-Francoise Roy
<span title="2009-11-06">2009</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove explicit bounds 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 quantifier-free formula involving s polynomials in Z[X_1, ..., X_k] having degrees at most d, and whose coefficients have bitsizes at most τ. Our bound is an explicit function of s, d, k and τ, and does not contain any undetermined constants. We also prove a similar bound on the radius of a ball guaranteed to
more &raquo; ... every connected component of S (including the unbounded components). While asymptotic bounds of the form 2^τ 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 τ. The bounds proved in this paper are of this nature.
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