We show that there is a defining equation of degree at most $\mathsf{poly}(n)$ for the (Zariski closure of the) set of the non-rigid matrices: that is, we show that for every large enough field $\mathbb{F}$, there is a non-zero $n^2$-variate polynomial $P \in \mathbb{F}[x_{1, 1}, \ldots, x_{n, n}]$ of degree at most $\mathsf{poly}(n)$ such that every matrix $M$ which can be written as a sum of a matrix of rank at most $n/100$ and a matrix of sparsity at most $n^2/100$ satisfies $P(M) = 0$. This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyer and Landsberg [GHIL16] and improves the best upper bound known for this problem down from $\exp(n^2)$ [KLPS14, GHIL16] to $\mathsf{poly}(n)$. We also show a similar polynomial degree bound for the (Zariski closure of the) set of all matrices $M$ such that the linear transformation represented by $M$ can be computed by an algebraic circuit with at most $n^2/200$ edges (without any restriction on the depth). As far as we are aware, no such bound was known prior to this work when the depth of the circuits is unbounded. Our methods are elementary and short and rely on a polynomial map of Shpilka and Volkovich [SV15] to construct low degree "universal" maps for non-rigid matrices and small linear circuits. Combining this construction with a simple dimension counting argument to show that any such polynomial map has a low degree annihilating polynomial completes the proof. As a corollary, we show that any derandomization of the polynomial identity testing problem will imply new circuit lower bounds. A similar (but incomparable) theorem was proved by Kabanets and Impagliazzo [KI04].

中文翻译：

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非刚性矩阵和小线性电路方程的多项式度界

我们证明，对于非刚性矩阵的（Zariski 闭包）集合，存在至多 $\mathsf{poly}(n)$ 的定义度方程：也就是说，我们证明对于每个足够大的域 $ \mathbb{F}$，存在一个非零 $n^2$-variate 多项式 $P \in \mathbb{F}[x_{1, 1}, \ldots, x_{n, n}]$度数至多 $\mathsf{poly}(n)$ 使得每个矩阵 $M$ 可以写成一个最多 $n/100$ 的秩矩阵和最多 $n^2 的稀疏矩阵的和/100$ 满足 $P(M) = 0$。这证实了 Gesmundo、Hauenstein、Ikenmeyer 和 Landsberg [GHIL16] 的猜想，并将此问题已知的最佳上限从 $\exp(n^2)$ [KLPS14, GHIL16] 改进为 $\mathsf{poly}(n )$。我们还展示了所有矩阵 $M$ 的（Zariski 闭包的）集合的相似多项式度界，使得 $M$ 表示的线性变换可以通过代数电路计算，最多为 $n^2/200$边缘（对深度没有任何限制）。据我们所知，当电路的深度是无界的时，在这项工作之前是不知道这样的界限的。我们的方法是基本的和简短的，依靠 Shpilka 和 Volkovich [SV15] 的多项式映射来构建非刚性矩阵和小型线性电路的低阶“通用”映射。将此构造与简单的维数计数参数相结合，以表明任何此类多项式映射具有低次湮灭多项式，从而完成了证明。作为推论，我们表明多项式身份测试问题的任何去随机化都将意味着新的电路下界。Kabanets 和 Impagliazzo [KI04] 证明了一个类似（但不可比）的定理。