  The relation between saddle points of the potential of a classical
many-particle system and the analyticity properties of its Boltzmann entropy is
studied. For finite systems, each saddle point is found to cause a
nonanalyticity in the Boltzmann entropy, and the functional form of this
nonanalytic term is derived for the generic case of potentials having the Morse
property. With increasing system size the order of the nonanalytic term grows
unboundedly, leading to an increasing differentiability of the entropy.
Nonetheless, a distribution of an unboundedly growing number of saddle points
may cause a phase transition in the thermodynamic limit. Analyzing the
contribution of the saddle points to the density of states in the thermodynamic
limit, conditions on the distribution of saddle points and their curvatures are
derived which are necessary for a phase transition to occur. With these
results, the puzzling absence of topological signatures in the spherical model
is elucidated. As further applications, the phase transitions of the mean-field
XY model and the mean-field k-trigonometric model are shown to be induced by
saddle points of vanishing curvature.
