Functional degrees and arithmetic applications III: Beyond Prime Exponent

Continuing our work on group-theoretic generalizations of the prime Ax-Katz Theorem, we give a lower bound on the $p$-adic divisibility of the cardinality of the set of simultaneous zeros $Z(f_1,f_2,\dots ,f_r)$ of $r$ maps $f_j:A\to B_j$ between arbitrary finite commutative groups $A$ and $B_j$ in terms of the invariant factors of $A,B_1,B_2,\dots ,B_r$ and the \emph{functional degrees} of the maps $f_1,f_2,\dots ,f_r$.