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We give a "regularity lemma" for degree-d polynomial threshold functions (PTFs) over the Boolean cube -1,1^n. This result shows that every degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a "regular PTF is a PTF sign(p(x)) where the influence of each variable on the polynomial p(x) is a small fraction of the total influence of p. As an application of this regularity lemma, we prove that for anyarXiv:0909.4727v2 fatcat:xdezqawno5d4jmq24wuritcvsu