Before revision:  The two characterizations , while in a certain sense equivalent, lead to two  different decision procedures  for checking whether a given semi-algebraic set is positively invariant under the flow of a system of polynomial ordinary differential equations  .
After revision:  The two characterizations , while in a certain sense equivalent, lead to two different decision procedures for checking whether a given semi-algebraic set is positively invariant under the flow of a system of polynomial ordinary differential equations . The procedure LZZ improves upon the original work by Liu, Zhan and Zhao (EMSOFT 2011). The procedure ESE, introduced in this article, works by splitting the problem, in a principled way, into simpler sub-problems that are easier to check, and is shown to exhibit substantially better performance compared to LZZ on problems featuring semi-algebraic sets described by formulas with non-trivial Boolean structure .