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On variety of monomials in a constrained symmetric polynomial Answer

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On variety of monomials in a constrained symmetric polynomial

Is there a symmetric polynomial $g$ having variety of monomials ${poly(n)}$ such that $$g(x_1,dots,x_n)equiv1bmod 2iff2sum_{i=1}^nx_igeq n$$ at each $(x_1,dots,x_n)in{0,1}^n$?

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