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 (termed respectively LZZ and ESE) for checking whether a given semi-algebraic set is positively invariant under the flow of a system of polynomial ordinary differential equations .