Everett W. Howe: Bounds on polarizations of abelian varieties over finite fields, J. Reine Angew. Math. 467 (1995) 149–155, MR 96i:11064, Zbl 0832.14033.
(An official version is available online.)
Using results from the preceding paper, I give explicit bounds for the degree of the smallest polarization of an abelian variety in a given isogeny class of varieties over a finite field. The bounds depend only on the dimension of the variety and the size of the base field. (Incidentally, in the published version of the paper, the name in the acknowledgments should be Hendrik Lenstra. Geez! So much for adding notes in proof!)