  The aim of this paper is to study the relationship between reduction numbers
and Borel-fixed ideals in all characteristics. By definition, Borel-fixed
ideals are closed under certain specializations which is similar to the strong
stability. We will estimate the number of monomials which can be specialized to
a given monomial. As a consequence, we obtain a combinatorial version of the
well-known Eakin-Sathaye's theorem which bounds the reduction number in terms
of the Hilbert function. Furthermore, we show that the bound of Eakin-Sathaye's
theorem is attained by the reduction number of a lex-segment monomial ideal.
This result answers a question of Conca in the affirmative. We will also show
that the reduction number of the lex-segment ideal is bounded exponentially by
the reduction number of the given ideal.
