  Geometry of sparse systems of polynomial equations (i.e. the ones with
prescribed monomials and generic coefficients) is well studied in terms of
their Newton polytopes. The results of this study are colloquially known as the
Bernstein--Kouchnirenko--Khovanskii toolkit, and unfortunately are not
applicable to many important systems, whose coefficients slightly fail to be
generic.
  This for instance happens if some of the equations are obtained from another
one by taking partial derivatives or permuting the variables, or the equations
are linear, realizing a non-trivial matroid, or in more advanced settings such
as generalized Calabi--Yau complete intersections.
  Such interesting examples (as well as many others) turn out to belong to a
natural class of ``systems of equations that are nondegenerate upon
cancellations''. We extend to this class several classical and folklore results
of the Bernstein--Kouchnirenko--Khovanskii toolkit, such as the ones regarding
the number and regularity of solutions, their irreducibility, tropicalization
and Calabi--Yau-ness.
