  The foundation of a matroid is a canonical algebraic invariant which
classifies representations of the matroid up to rescaling equivalence.
Foundations of matroids are pastures, a simultaneous generalization of partial
fields and hyperfields.
  Using deep results due to Tutte, Dress-Wenzel, and Gelfand-Rybnikov-Stone, we
give a presentation for the foundation of a matroid in terms of generators and
relations. The generators are certain "cross-ratios" generalizing the
cross-ratio of four points on a projective line, and the relations encode
dependencies between cross-ratios in certain low-rank configurations arising in
projective geometry.
  Although the presentation of the foundation is valid for all matroids, it is
simplest to apply in the case of matroids without large uniform minors, i.e.
matroids having no minor corresponding to five points on a line or its dual
configuration. For such matroids, we obtain a complete classification of all
possible foundations. We then give a number of applications of this
classification theorem, for example:
  - We prove the following strengthening of a theorem of Lee and Scobee: every
orientation of a matroid without large uniform minors comes from a dyadic
representation, which is unique up to rescaling.
  - For a matroid $M$ without large uniform minors, we establish the following
strengthening of a 2017 theorem of Ardila-Rinc\'on-Williams: if $M$ is
positively oriented then $M$ is representable over every field with at least
three elements.
  - Two matroids are said to belong to the same representation class if they
are representable over precisely the same pastures. We prove that there are
precisely 12 possibilities for the representation class of a matroid without
large uniform minors, exactly three of which are not representable over any
field.
