  Given a set of inequalities determined by homogeneous forms, the following
intertwined results are established:
  (1) the volume of the real semi-algebraic domain determined by these
inequalities is explicitly determined; it is shown to be related to the largest
root of the so-called Sato-Bernstein polynomial associated to a multivarite
polynomial derived from the given set of homogeneous forms;
  (2) in relation with this result, the multiplicity of the largest root of the
Sato-Bernstein polynomial of a multivariate polynomial is shown to coincide
with the order of the smallest pole of the complex meromorphic
zeta-distribution attached to it. This settles a classical problem in the
theory of D-modules;
  (3) in the case that the homogeneous forms are twisted by random unimodular
matrices, a metric, uniform and effective version of the Oppenheim conjecture
is established. This answers a problem raised by Athreya and Margulis (2018).
So does a related metric estimate counting the number of solutions in integer
lattice points to the set of twisted inequalities.
  (4) in the deterministic case where the set of homogeneous forms is fixed, an
upper bound is proved to hold for the function counting the number of integer
solutions to the system of inequalities under consideration. The error term in
this estimate is shown to admit a power saving provided that a quantitative
measure of flatness emerging from geometric tomography is large enough. This
settles a conjecture stated by Sarnak (1997).

Number Theory;Algebraic Geometry;Logic