  Area variables are intrinsic to connection formulations of general
relativity, in contrast to the fundamental length variables prevalent in metric
formulations. Within 4D discrete gravity, particularly based on triangulations,
the area-length system establishes a relationship between area variables
associated with triangles and the edge length variables. This system is
comprised of polynomial equations derived from Heron's formula, which relates
the area of a triangle to its edge lengths.
  Using tools from numerical algebraic geometry, we study the area-length
systems. In particular, we show that given the ten triangular areas of a single
4-simplex, there could be up to 64 compatible sets of edge lengths. Moreover,
we show that these 64 solutions do not, in general, admit formulae in terms of
the areas by analyzing the Galois group, or monodromy group, of the problem. We
show that by introducing additional symmetry constraints, it is possible to
obtain such formulae for the edge lengths. We take the first steps toward
applying our results within discrete quantum gravity, specifically for
effective spin foam models.
