We give a "regularity lemma" for degree-d polynomial threshold functions (PTFs) over the Boolean cube {−1,1}. Roughly speaking, 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 any constants d≥1,ϵ>0, every degree-d PTF over n variables can be approximated to accuracy ϵ by a constant-degree PTF that has integer weights of total magnitude O
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