On the minimum of a polynomial function on a basic closed semialgebraic   set and applications
  We give an explicit upper bound for the algebraic degree and an explicit lower bound for the absolute value of the minimum of a polynomial function on a compact connected component of a basic closed semialgebraic set when this minimum is not zero. As an application, we obtain a lower bound for the separation of two disjoint connected components of basic closed semialgebraic sets, when at least one of them is compact. 
Gabriela Jeronimo, Daniel Perrucci, Elias Tsigaridas
math.AG cs.CG