Lower Bounds on Representing Boolean Functions as Polynomials in Zm
نویسنده
چکیده
Deene the MOD m-degree of a boolean function F to be the smallest degree of any polynomial P, over the ring of integers modulo m, such that for all 0-1 assignments ~ x, F(~ x) = 0 ii P(~ x) = 0. By exploring the periodic property of the binomial coeecients modulo m, two new lower bounds on the MOD m-degree of the MOD l and :MOD m functions are proved, where m is any composite integer and l has a prime factor not dividing m. Both bounds improve from sub-linear to (n). With the periodic property, a simple proof of a lower bound on the MOD m-degree with symmetric multilinear polynomial of the OR function is given. It is also proved that the majority function has a lower bound n 2 and the MidBit function has a lower bound p n.
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