A Degree-Decreasing Lemma for (MOD q - MOD p) Circuits

Consider a (MODq;MODp) circuit, where the inputs of the bottom MODp gates are degree-d polynomials with integer coefficients of the input variables (p, q are different primes). Using our main tool — the Degree Decreasing Lemma — we show that this circuit can be converted to a (MODq;MODp) circuit with linear polynomials on the input-level with the price of increasing the size of the circuit. This result implies special cases of the Constant Degree Hypothesis of Barrington, Straubing and Thérien [3], and implies also a generalization of the lower bound results of Yan and Parberry [21], Krause and Waack [12] and Krause and Pudlák [11]. Perhaps the most important application is an exponential lower bound for the size of (MODq;MODp) circuits computing the fan-in n AND, where the input of each MODp gate at the bottom is an arbitrary integer valued function of cn variables (c < 1) plus an arbitrary linear function of n input variables.