  Symanzik polynomials are defined on Feynman graphs and they are used in
quantum field theory to compute Feynman amplitudes. They also appear in
mathematics from different perspectives. For example, recent results show that
they allow to describe asymptotic limits of geometric quantities associated to
families of Riemann surfaces.
  In this paper, we propose a generalization of Symanzik polynomials to the
setting of higher dimensional simplicial complexes and study their basic
properties and applications. We state a duality relation between these
generalized Symanzik polynomials and what we call Kirchhoff polynomials, which
have been introduced in recent generalizations of Kirchhoff's matrix-tree
theorem to simplicial complexes. Moreover, we obtain geometric invariants which
compute interesting data on triangulable manifolds. As the name indicates,
these invariants do not depend on the chosen triangulation.
  We furthermore prove a stability theorem concerning the ratio of Symanzik
polynomials which extends a stability theorem of Amini to higher dimensional
simplicial complexes. In order to show that theorem, we will make great use of
matroids, and provide a complete classification of the connected components of
the exchange graph of a matroid, a graph which encodes the exchange properties
between independent sets of the matroid. We hope that this result could be of
independent interest.
  Finally, we explain how to generalize Symanzik polynomials to the setting of
matroids over hyperfields defined recently by Baker and Bowler.
