  Scattering amplitudes in quantum field theories have intricate analytic
properties as functions of the energies and momenta of the scattered particles.
In perturbation theory, their singularities are governed by a set of nonlinear
polynomial equations, known as Landau equations, for each individual Feynman
diagram. The singularity locus of the associated Feynman integral is made
precise with the notion of the Landau discriminant, which characterizes when
the Landau equations admit a solution. In order to compute this discriminant,
we present approaches from classical elimination theory, as well as a numerical
algorithm based on homotopy continuation. These methods allow us to compute
Landau discriminants of various Feynman diagrams up to 3 loops, which were
previously out of reach. For instance, the Landau discriminant of the envelope
diagram is a reducible surface of degree 45 in the three-dimensional space of
kinematic invariants. We investigate geometric properties of the Landau
discriminant, such as irreducibility, dimension and degree. In particular, we
find simple examples in which the Landau discriminant has codimension greater
than one. Furthermore, we describe a numerical procedure for determining which
parts of the Landau discriminant lie in the physical regions. In order to study
degenerate limits of Landau equations and bounds on the degree of the Landau
discriminant, we introduce Landau polytopes and study their facet structure.
Finally, we provide an efficient numerical algorithm for the computation of the
number of master integrals based on the connection to algebraic statistics. The
algorithms used in this work are implemented in the open-source Julia package
Landau.jl available at https://mathrepo.mis.mpg.de/Landau/.
