We show that any Algebraic Branching Program (ABP) computing the polynomial $\sum_{i = 1}^n x_i^n$ has at least $\Omega(n^2)$ vertices. This improves upon the lower bound of $\Omega(n\log n)$, which follows from the classical result of Baur and Strassen [Str73, BS83], and extends the results in [K19], which showed a quadratic lower bound for \emph{homogeneous} ABPs computing the same polynomial. Our proof relies on a notion of depth reduction which is reminiscent of similar statements in the context of matrix rigidity, and shows that any small enough ABP computing the polynomial $\sum_{i=1}^n x_i^n$ can be depth reduced to essentially a homogeneous ABP of the same size which computes the polynomial $\sum_{i = 1}^n x_i^n + \epsilon(x_1, \ldots, x_n)$, for a structured "error polynomial" $\epsilon(x_1, \ldots, x_n)$. To complete the proof, we then observe that the lower bound in [K19] is robust enough and continues to hold for all polynomials $\sum_{i = 1}^n x_i^n + \epsilon(x_1, \ldots, x_n)$, where $\epsilon(x_1, \ldots, x_n)$ has the appropriate structure. We also use our ideas to show an $\Omega(n^2)$ lower bound of the size of algebraic formulas computing the elementary symmetric polynomial of degree $0.1n$ on $n$ variables. This is a slight improvement upon the prior best known formula lower bound (proved for a different polynomial) of $\Omega(n^2/\log n)$ [Nec66, K85, SY10]. Interestingly, this lower bound is asymptotically better than $n^2/\log n$, the strongest lower bound that can be proved using previous methods. This lower bound also matches the upper bound, due to Ben-Or, who showed that elementary symmetric polynomials can be computed by algebraic formula (in fact depth-$3$ formula) of size $O(n^2)$. Prior to this work, Ben-Or's construction was known to be optimal only for algebraic formulas of depth-$3$ [SW01].

中文翻译：

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代数分支程序和公式的二次下界

我们表明，任何计算多项式 $\sum_{i = 1}^n x_i^n$ 的代数分支程序 (ABP) 至少有 $\Omega(n^2)$ 个顶点。这改进了 $\Omega(n\log n)$ 的下界，该下界遵循 Baur 和 Strassen [Str73, BS83] 的经典结果，并扩展了 [K19] 中的结果，该结果显示出二次下界\emph{homogeneous} ABP 计算相同的多项式。我们的证明依赖于深度减少的概念，这让人想起矩阵刚性上下文中的类似陈述，并表明任何足够小的 ABP 计算多项式 $\sum_{i=1}^n x_i^n$ 都可以深度减少本质上是一个相同大小的齐次 ABP，它计算多项式 $\sum_{i = 1}^n x_i^n + \epsilon(x_1, \ldots, x_n)$，​​对于结构化的“误差多项式” $\epsilon( x_1, \ldots, x_n)$。为了完成证明，然后我们观察到 [K19] 中的下界足够健壮，并且继续适用于所有多项式 $\sum_{i = 1}^n x_i^n + \epsilon(x_1, \ldots, x_n)$，​​其中 $\ epsilon(x_1, \ldots, x_n)$ 具有适当的结构。我们还使用我们的想法来显示代数公式大小的 $\Omega(n^2)$ 下界，用于计算 $n$ 变量上的 $0.1n$ 次基本对称多项式。这是对 $\Omega(n^2/\log n)$ [Nec66, K85, SY10] 的先前最知名公式下界（针对不同的多项式证明）的略微改进。有趣的是，这个下界渐近优于 $n^2/\log n$，这是可以使用以前的方法证明的最强下界。由于 Ben-Or，此下限也与上限匹配，谁表明基本对称多项式可以通过大小为 $O(n^2)$ 的代数公式（实际上是 depth-$3$ 公式）计算。在此工作之前，已知 Ben-Or 的构造仅适用于深度为 $3$ [SW01] 的代数公式。