  We reformulate the Landau analysis of Feynman integrals with the aim of
advancing the state of the art in modern particle-physics computations. We
contribute new algorithms for computing Landau singularities, using tools from
polyhedral geometry and symbolic/numerical elimination. Inspired by the work of
Gelfand, Kapranov, and Zelevinsky (GKZ) on generalized Euler integrals, we
define the principal Landau determinant of a Feynman diagram. We illustrate
with a number of examples that this algebraic formalism allows to compute many
components of the Landau singular locus. We adapt the GKZ framework by
carefully specializing Euler integrals to Feynman integrals. For instance,
ultraviolet and infrared singularities are detected as irreducible components
of an incidence variety, which project dominantly to the kinematic space. We
compute principal Landau determinants for the infinite families of one-loop and
banana diagrams with different mass configurations, and for a range of
cutting-edge Standard Model processes. Our algorithms build on the Julia
package Landau.jl and are implemented in the new open-source package PLD.jl
available at https://mathrepo.mis.mpg.de/PLD/.
