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nt.quantity principle – On variety of monomials in inescapable constrained polynomials Answer

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nt.quantity principle – On variety of monomials in inescapable constrained polynomials

At each $nin{1,2,dots,}$ and at each $qinmathcal I={2,3,dots,p}$ is there a polynomial $g_q$ the place $mathcal I$ is a clique of primes and $p=O(1)$ having variety of monomials ${poly(n)}$ satisfying the circumstances

  1. $$Big(oplus_{qinmathcal I}mathbb 1_{g_q(x_1,dots,x_n)bmod q}Big)equiv1iff2sum_{i=1}^nx_igeq n$$
    at each $(x_1,dots,x_n)in{0,1}^n$ the place $mathbb 1_a=1iff a=0$ and the place $oplus$ is $bmod 2$

  2. $g_q(x_1,dots,x_n)inmathbb Z[x_1,dots,x_n]$

  3. each coefficient of $g_q$ has magnitude sure by $2^{poly(n)}$

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