For a Z_p^d-tower of a global field K, Greenberg conjectured
that the exponent of p dividing the class number of the n-th
extension in the tower is given by a polynomial in n and p^n of total degree at
most d for all large n. This conjecture is well known to be true for d=1
(classical Iwasawa theory) but remains open for $d>1$. In this talk,
we explain why the conjecture is true when K is a global function field.