The sheaf-function dictionary shows that many natural functions on the F_q-points of a variety over F_q can be obtained from l-adic sheaves on that variety. To obtain upper bounds on these functions, it is necessary to obtain upper bounds on the dimension of the stalks of these sheaves. In many number theory problems, the functions we need to bound arise from perverse sheaves. David Massey proved an upper bound for the dimension of a stalk of a perverse sheaf in terms of an intersection-theoretic invariant of the characteristic cycle called a polar multiplicity. Unfortunately, he proved this only over the complex numbers, making it unusable for these problems. I will explain how to rectify this by proving an analogous statement over finite fields. This has applications to function field analogues of three problems in number theory: The Michel-Venkatesh conjecture about equidistribution of CM points, a question about large values of automorphic forms, and the number of primes in an arithmetic progression.