  One of the general problems in algebraic geometry is to determine
algorithmically whether or not a given geometric object, defined by explicit
polynomial equations (e.g. a curve or a surface), satisfies a given property
(e.g. has singularities or other distinctive features of interest). A classical
example of such a problem, described by A.Cayley and G.Salmon in 1852, is to
determine whether or not a given plane curve of degree r > 3 has undulation
points -- the points where the tangent line meets the curve with multiplicity
four. They proved that there exists an invariant of degree 6(r - 3)(3 r - 2)
that vanishes if and only if the curve has undulation points. In this paper we
give explicit formulae for this invariant in the case of quartics (r=4) and
quintics (r=5), expressing it as the determinant of a matrix with polynomial
entries, of sizes 21 times 21 and 36 times 36 respectively.
