  This paper is a systematic study of the Hilbert polynomial of a bigraded
algebra R which are generated by elements of bidegrees (1,0),
(d_1,1),...,(d_r,1), where d_1,...,d_r are non-negative integers. The obtained
results can be applied to study Rees algebras of homogeneous ideals and their
diagonal subalgebras. The Hilbert polynomial is computed explicitly in the
following cases: (1) R is a bigraded polynomial ring of the above type; (2) R
is the Rees algebra of an ideal generated by a homogeneous regular sequence;
(3) R is the Rees algebra of the ideal generated by the maximal minors of a
generic (r-1) by r matrix.
