On representations by low-degree polynomials

In the first part of the paper we show that a subset S of a boolean cube B/sub n/ embedded in the projective space P/sup n/ can be approximated by a subset of B/sub n/ defined by nonzeroes of a low-degree polynomial only if the values of the Hilbert function of S are sufficiently small relative to the size of S. The use of this property provides a simple and direct technique for proving lower bounds on the size of ACC[p/sup r/] circuits. In the second part we look at the problem of computing many-output function by ACC[p/sup r/] circuit and give an example when such a circuit can be correct only at exponentially small fraction of assignments.<
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