  We extend a result of Caviglia and Sbarra to a polynomial ring with base
field of any characteristic. Given a homogeneous ideal containing both a
piecewise lex ideal and an ideal generated by powers of the variables, we find
a lex ideal with the following property: the ideal in the polynomial ring
generated by the piecewise lex ideal, the ideal of powers, and the lex ideal
has the same Hilbert function and Betti numbers at least as large as those of
the original ideal.
