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Consider a (MOD_q,MOD_p) circuit, where the inputs of the bottom MOD_p 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 (MOD_q,MOD_p) circuit with \emphlinear polynomials on the input-level with the price of increasing the size of the circuit. This result has numerous consequences: for the Constant Degree Hypothesis of Barrington, Straubing and Thérien, and generalizing the lower bound results of Yan and Parberry, Krause and Waack, and Krause and Pudlák. Perhaps the most important application is an exponential lower bound for the size of (MOD_q,MOD_p) circuits computing the n fan-in AND, where the input of each MOD_p gate at the bottom is an \empharbitrary integer valued function of cn variables (c&lt;1) plus an arbitrary linear function of n input variables.

A Degree-Decreasing Lemma for (MOD_q-MOD_p) Circuits