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Algebraic independence over positive characteristic: New criterion and applications to locally low-algebraic-rank circuits
computational complexity ( IF 1.4 ) Pub Date : 2018-05-14 , DOI: 10.1007/s00037-018-0167-5
Anurag Pandey , Nitin Saxena , Amit Sinhababu

The motivation for this work (Pandey et al. 2016) comes from two problems: testing algebraic independence of arithmetic circuits over a field of small characteristic and generalizing the structural property of algebraic dependence used by Kumar, Saraf, CCC’16 to arbitrary fields. It is known that in the case of zero, or large characteristic, using a classical criterion based on the Jacobian, we get a randomized poly-time algorithm to test algebraic independence. Over small characteristic, the Jacobian criterion fails and there is no subexponential time algorithm known. This problem could well be conjectured to be in RP, but the current best algorithm puts it in NP$${^{\#{\rm P}}}$$#P (Mittmann, Saxena, Scheiblechner, Trans.AMS’14). Currently, even the case of two bivariate circuits over $${\mathbb{F}_2}$$F2 is open. We come up with a natural generalization of Jacobian criterion that works over all characteristics. The new criterion is efficient if the underlying inseparable degree is promised to be a constant. This is a modest step toward the open question of fast independence testing, over finite fields, posed in (Dvir, Gabizon, Wigderson, FOCS’07). In a set of linearly dependent polynomials, any polynomial can be written as a linear combination of the polynomials forming a basis. The analogous property for algebraic dependence is false, but a property approximately in that spirit is named as “functional dependence” in Kumar, Saraf, CCC’16 and proved for zero or large characteristics. We show that functional dependence holds for arbitrary fields, thereby answering the open questions in Kumar, Saraf, CCC’16. Following them, we use the functional dependence lemma to prove the first exponential lower bound for locally low algebraic rank circuits for arbitrary fields (a model that strongly generalizes homogeneous depth-4 circuits). We also recover their quasipoly-time hitting-set for such models, for fields of characteristic smaller than the ones known before. Our results show that approximate functional dependence is indeed a more fundamental concept than the Jacobian as it is field independent. We achieve the former by first picking a “good” transcendence basis, then translating the circuits by new variables, and finally approximating them by truncating higher degree monomials. We give a tight analysis of the “degree” of approximation needed in the criterion. To get the locally low-algebraic-rank circuit applications, we follow the known shifted partial derivative-based methods.

中文翻译:

正特性上的代数独立性:局部低代数秩电路的新判据和应用

这项工作(Pandey 等人,2016 年)的动机来自两个问题:在小特征域上测试算术电路的代数独立性,以及将 Kumar、Saraf、CCC'16 使用的代数依赖的结构特性推广到任意域。众所周知,在零或大特征的情况下,使用基于雅可比的经典准则,我们得到一个随机的多时间算法来测试代数独立性。在小特征上,雅可比准则失败并且没有已知的次指数时间算法。这个问题很可能被推测出在 RP 中,但目前最好的算法把它放在 NP$${^{\#{\rm P}}}$$#P (Mittmann, Saxena, Scheiblechner, Trans.AMS'14 )。目前,即使是 $${\mathbb{F}_2}$$F2 上的两个双变量电路的情况也是开放的。我们提出了适用于所有特征的雅可比准则的自然概括。如果潜在的不可分度承诺为常数,则新标准是有效的。这是朝着在 (Dvir, Gabizon, Wigderson, FOCS'07) 中提出的有限域上的快速独立性测试这一开放性问题迈出的适度的一步。在一组线性相关多项式中,任何多项式都可以写成多项式的线性组合,形成一个基。代数依赖的类似性质是错误的,但近似于这种精神的性质在 Kumar、Saraf、CCC'16 中被命名为“函数依赖”,并证明了零或大特征。我们证明函数依赖适用于任意领域,从而回答了 Kumar、Saraf、CCC'16 中的开放性问题。跟着他们,我们使用函数依赖引理来证明任意域的局部低代数秩电路的第一个指数下界(一个强烈推广同构深度 4 电路的模型)。我们还为这些模型恢复了它们的准聚时间命中集,对于特征域比以前已知的域更小。我们的结果表明近似函数依赖确实是一个比雅可比矩阵更基本的概念,因为它是场独立的。我们通过首先选择一个“好的”超越基,然后用新变量转换电路,最后通过截断更高次的单项式来逼近它们来实现前者。我们对标准中所需的近似“程度”进行了严格的分析。为了获得局部低代数秩电路应用,
更新日期:2018-05-14
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