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In this paper, we characterize the (generalized) Frobenius powers andcritical exponents of two classes of monomial ideals of a polynomial ring inpositive characteristic: powers of the homogeneous maximal ideal, and idealsgenerated by positive powers of the variables. In doing so, we effectivelycharacterize the test ideals and $F$-jumping exponents of sufficiently generalhomogeneous polynomials, and of all diagonal polynomials. Our characterizationsmake these invariants computable, and show that they vary uniformly with thecongruence class of the characteristic modulo a fixed integer. Moreover, weconfirm that for a diagonal polynomial over a field of characteristic zero, thetest ideals of its reduction modulo a prime agree with the reductions of itsmultiplier ideals for infinitely many primes.

Frobenius powers of some monomial ideals