  Let f be a polynomial of degree n in ZZ[x_1,..,x_n], typically reducible but
squarefree. From the hypersurface {f=0} one may construct a number of other
subschemes {Y} by extracting prime components, taking intersections, taking
unions, and iterating this procedure.
  We prove that if the number of solutions to f=0 in \FF_p^n is not a multiple
of p, then all these intersections in \AA^n_{\FF_p} just described are reduced.
(If this holds for infinitely many p, then it holds over \QQ as well.) More
specifically, there is a_Frobenius splitting_ on \AA^n_{\FF_p} compatibly
splitting all these subschemes {Y}.
  We determine when a Gr\"obner degeneration f_0=0 of such a hypersurface f=0
is again such a hypersurface. Under this condition, we prove that compatibly
split subschemes degenerate to compatibly split subschemes, and stay reduced.
  Our results are strongest in the case that f's lexicographically first term
is \prod_{i=1}^n x_i. Then for all large p, there is a Frobenius splitting that
compatibly splits f's hypersurface and all the associated {Y}. The Gr\"obner
degeneration Y' of each such Y is a reduced union of coordinate spaces (a
Stanley-Reisner scheme), and we give a result to help compute its Gr\"obner
basis. We exhibit an f whose associated {Y} include Fulton's matrix Schubert
varieties, and recover much more easily the Gr\"obner basis theorem of
[Knutson-Miller '05]. We show that in Bott-Samelson coordinates on an opposite
Bruhat cell X^v_\circ in G/B, the f defining the complement of the big cell
also has initial term \prod_{i=1}^n x_i, and hence the Kazhdan-Lusztig
subvarieties {X^v_{w\circ}} degenerate to Stanley-Reisner schemes. This
recovers, in a weak form, the main result of [Knutson '08].
