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 = \binom{n^c + n}{n}, P_{N,c} \emph{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 = \binom{n^c + n}{n}, P_{N,c} \emph{does not vanish} on the coefficient vector of h_n. In other words, there are efficiently computable 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 \emph{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 equations for these classes. Our proofs also extend to finite fields of small size.
翻译:对于每个常数 c > 0, 我们显示, 在 f_ n 的变量中, 有家庭 {P{N}, c ⁇ 的多球体, 其程度和代数电路复杂度在变量数量上是多元的, 这满足了以下属性 : * 对于 VP 中每个多球体 {f_n} 的家庭 {f_n}, 其中f_n 最多为Nvval- 多元度, 最多为nvc, 其中F_n=binom{n{c + n ⁇ n}, N=\binom{n{n}, p ⁇ {c+n}, P{N} 等离子体( listal), P{nph{vanishes} 在 f_n_n 的变数矢量矢量矢量中, 存在一个家族 {h_n_n_n} 的多球体系, 其中h_n=nvlational- mexal log log 。