On the Existence of Algebraically Natural Proofs
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For every constant c > 0, we show that there is a family {P_N, c} of polynomials whose degree and algebraic circuit complexity are polynomially bounded in the number of variables, that satisfies the following properties: ∙ For every family {f_n} of polynomials in VP, where f_n is an n variate polynomial of degree at most n^c with bounded integer coefficients and for N = n^c + nn, P_N,c vanishes on the coefficient vector of f_n. ∙ There exists a family {h_n} of polynomials where h_n is an n variate polynomial of degree at most n^c with bounded integer coefficients such that for N = n^c + nn, P_N,c does not vanish on the coefficient vector of h_n. In other words, there are efficiently computable defining equations for polynomials in VP that have small integer coefficients. In fact, we also prove an analogous statement for the seemingly larger class VNP. Thus, in this setting of polynomials with small integer coefficients, this provides evidence against a natural proof like barrier for proving algebraic circuit lower bounds, a framework for which was proposed in the works of Forbes, Shpilka and Volk (2018), and Grochow, Kumar, Saks and Saraf (2017). Our proofs are elementary and rely on the existence of (non-explicit) hitting sets for VP (and VNP) to show that there are efficiently constructible, low degree defining equations for these classes and also extend to finite fields of small size.
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