  Exploiting tools from algebraic geometry, the problem of finiteness of
determination of accessibility/strong accessibility is investigated for
polynomial systems and also for analytic systems that are immersible into
polynomial systems. The results are constructive, and algorithms are given to
find the maximum depth of Lie brackets necessary for deciding
accessibility/strong accessibility of the system at any point, called here
accessibility/strong accessibility index of the system, and is known as the
degree of non-holonomy in the literature. Alternatively, upper bounds on the
accessibility/strong accessibility index are obtained, which can be computed
easier. In each approach, the entire set of accessibility/strong accessibility
singular points are obtained.
