On the Existence of Algebraically Natural Proofs
release_qsxtdunpszfjjjss62hpr7orjq
by
Prerona Chatterjee, Mrinal Kumar, C. Ramya, Ramprasad Saptharishi, Anamay Tengse
2020
Abstract
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,cvanishes 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,cdoes 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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