发现根：代数类在因子分解下的一致闭包结果
Discovering the roots: Uniform closure results for algebraic classes under factoring
摘要：
牛顿迭代（NI）是一个有着近350年历史的递归公式，它可以非常快速地近似多项式的简单根。我们将其推广到同时逼近所有根的矩阵递归（allRootsNI）。在这种形式中，当根$r$的数量很小但多重性指数很大时，该过程产生更好的电路复杂度。我们的方法在$r$未知数中建立一个线性系统，并迭代地将根建立为形式幂级数。对于一个代数电路$f(x_1,\ldots,x_n)$的大小$s$，我们证明了每个因素的大小最多一个多项式：$s$和程度的平方部分$f$。因此，如果$f_1$是一个$2^{Ω(n)}$ -硬多项式，那么任何非零的多个$\prod_{i} f_i^{e_i}$对于任意正的$e_i$都是同样硬的，假设$\sum_i \text{deg}(f_i)$至多是$2^{O(n)}$ .这是一个古老的开放性问题，无论是聚（$n$）大小的公式类（分别。代数分支程序）在因子分解下是封闭的。我们证明了给定一个次数为$n^{O(1)}$的多项式$f$和公式（分别为. ABP）大小$n^{O(\log n)}$我们可以找到一个类似的大小公式（分别为。随机聚（$n^{\log n}$）-时间中的ABP）因子。因此，如果行列式需要$n^{Ω(\log n)}$大小公式，那么它的任何非零倍数也可以这样说。作为我们的证明的一部分，我们确定了多元多项式分解的一个新的属性。我们表明，在随机线性变换$τ$，$f(τ\overline{x})$完全因素通过幂级数根。此外，因式分解很好地适应电路的复杂性分析。这与allRootsNI是技术，帮助我们取得进展，对旧的开放问题，补充了大量的经典结果和概念，在代数电路因式分解（如。Zassenhaus，J.NT 1969，Kaltofen，STOC 1985-7 \& Burgisser，FOCS 2001）。
英文摘要：
Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots $r$ is small but the multiplicities are exponentially large. Our method sets up a linear system in $r$ unknowns and iteratively builds the roots as formal power series. For an algebraic circuit $f(x_1,\ldots,x_n)$ of size $s$ we prove that each factor has size at most a polynomial in: $s$ and the degree of the squarefree part of $f$. Consequently, if $f_1$ is a $2^{Ω(n)}$-hard polynomial then any nonzero multiple $\prod_{i} f_i^{e_i}$ is equally hard for arbitrary positive $e_i$'s, assuming that $\sum_i \text{deg}(f_i)$ is at most $2^{O(n)}$. It is an old open question whether the class of poly($n$)-sized formulas (resp. algebraic branching programs) is closed under factoring. We show that given a polynomial $f$ of degree $n^{O(1)}$ and formula (resp. ABP) size $n^{O(\log n)}$ we can find a similar size formula (resp. ABP) factor in randomized poly($n^{\log n}$)-time. Consequently, if determinant requires $n^{Ω(\log n)}$ size formula, then the same can be said about any of its nonzero multiples. As part of our proofs, we identify a new property of multivariate polynomial factorization. We show that under a random linear transformation $τ$, $f(τ\overline{x})$ completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. This with allRootsNI are the techniques that help us make progress towards the old open problems, supplementing the large body of classical results and concepts in algebraic circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \& Burgisser, FOCS 2001).
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