On the concentration of multivariate polynomials with small expectation

Let t1,…,tn be independent, but not necessarily identical, {0, 1} random variables. We prove a general large deviation bound for multivariate polynomials (in t1,…,tn) with small expectation [order O(polylog(n))]. Few applications in random graphs and combinatorial number theory will be discussed. Our result is closely related to a classical result of Janson [Random Struct Algorithms 1 (1990), 221–230]. Both of them can be applied in similar situations. On the other hand, our result is symmetric, while Janson's inequality only deals with the lower tail probability.