Properties of Lyubeznik numbers under localization and polarization

We exhibit a global bound for the Lyubeznik numbers of a ring of prime characteristic. In addition, we show that for a monomial ideal, the Lyubeznik numbers of the quotient rings of its radical and its polarization are the same. Furthermore, we present examples that show striking behavior of the Lyubeznik numbers under localization. We also show related results for generalizations of the Lyubeznik numbers.