On Polynomial Representations of Boolean Functions Related to Some Number Theoretic Problems
FST TCS '01 Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science
Combinatorics, Probability and Computing
Symmetric polynomials over Zm and simultaneous communication protocols
Journal of Computer and System Sciences - Special issue on FOCS 2003
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Define 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 $\vec{x}$, $F(\vec{x}) =0$ iff $P(\vec{x}) =0$. By exploring the periodic property of the binomial coefficients modulo $m$, two new lower bounds on the $MOD_{m}$-degree of the $MOD_{l}$ and $\neg MOD_{m}$ functions are proved, where $m$ is any composite integer and $l$ has a prime factor not dividing $m$. Both bounds improve from sublinear to $\Omega(n)$. 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 $n \over 2$ and the {MidBit} function has a lower bound $\sqrt{n}$.