  Let f(x) = f(x_1, ..., x_n) = \sum_{|S| <= k} a_S \prod_{i \in S} x_i be an
n-variate real multilinear polynomial of degree at most k, where S \subseteq
[n] = {1, 2, ..., n}. For its "one-block decoupled" version,
  f~(y,z) = \sum_{|S| <= k} a_S \sum_{i \in S} y_i \prod_{j \in S\i} z_j,
  we show tail-bound comparisons of the form
  Pr[|f~(y,z)| > C_k t] <= D_k Pr[f(x) > t].
  Our constants C_k, D_k are significantly better than those known for "full
decoupling". For example, when x, y, z are independent Gaussians we obtain C_k
= D_k = O(k); when x, y, z, Rademacher random variables we obtain C_k = O(k^2),
D_k = k^{O(k)}. By contrast, for full decoupling only C_k = D_k = k^{O(k)} is
known in these settings.
  We describe consequences of these results for query complexity (related to
conjectures of Aaronson and Ambainis) and for analysis of Boolean functions
(including an optimal sharpening of the DFKO Inequality).
