TITLE: Monte-Carlo computation of the Smith normal form
QUESTION [6 upvotes]: Quite some time ago I saw an article where a Monte-Carlo algorithm for computing the Smith normal form of an integer matrix was described. In this article the following problem was posed:
Suppose $P, Q\in\mathbb{Z}[X,Y]$ are polynomials without a common factor. Then $(P(n,m), Q(n,m))$ has a density, i.e. there exist real numbers $\delta_k$, such that
$$
\lim_{N\rightarrow\infty} \frac{1}{4N^2}\#\big\{(n,m):-N\leq n,m\leq N, (P(n,m), Q(n,m)) = k\big\} = \delta_k,
$$
and $\sum_k\delta_k = 1$.
Can someone please give me a reference to this paper?

REPLY [2 votes]: Typing "Monte-Carlo" and "Smith normal" into MathSciNet turned up a few possibilities. 
Mustafa Elsheikh, Mark Giesbrecht, Andy Novocin, B. David Saunders, Fast computation of Smith forms of sparse matrices over local rings, ISSAC 2012—Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, 146–153, ACM, New York, 2012, MR3206298. 
David Saunders, Zhendong Wan, Smith normal form of dense integer matrices, fast algorithms into practice. (English summary) ISSAC 2004, 274–281, ACM, New York, 2004, MR2126954 (2005k:15039). 
[The review mentions a "fast Monte Carlo algorithm of Eberly, Giesbrecht and Villard for computing the Smith form of an integer matrix," but gives no reference.]
Mark Giesbrecht, Fast computation of the Smith form of a sparse integer matrix, Comput. Complexity 10 (2001), no. 1, 41–69, MR1867308 (2003d:15014).