基于和集扩张的算术电路下界
Arithmetic circuit lower bounds from sumset expansion
AI总结：
研究通过构造难以捉摸的函数证明算术电路下界。新方法将坐标映射限于单项式，利用和集扩张及切博塔廖夫定理分析构造，解决相关开放问题，改进拉兹超线性界，为算术电路下界研究提供新思路。
AI中文摘要：
拉兹提出了一个通过明确构造难以捉摸的函数来证明算术电路下界的计划。这些函数是从低维空间到高维环境空间的多项式映射，其图像不包含低复杂度的子簇。难以捉摸的函数很丰富，但尚无已知的明确构造。拉兹设计了参数较弱的难以捉摸的函数，得到了深度\(d = o(\log n)\)时需要超线性电路规模的\(n\)元\(d\)次显式多项式。我们提出了一种新方法来分析和构造难以捉摸的函数，坐标映射限于单项式。为证明其难以捉摸性，我们确定了一组点的击中集，利用切博塔廖夫关于单位根的定理表明，对于每个低复杂度子簇，函数在击中集的某点求值时会避开它。这一策略奏效的条件是某组数字（由指数导出）的迭代和集呈指数扩张。我们将难以捉摸函数中的开放显式构造问题简化为纯加法组合问题，其解决意味着未知的下界。受迭代和集扩张启发，我们设计了新的难以捉摸的函数，构造了指数次数的显式难以捉摸曲线，解决了加尔格、马卡姆、奥利维拉和维格森提出的一个开放问题，并在深度低于\(o(\log n / \log\log n)\)时将拉兹的超线性界二次改进。
英文摘要：
Raz proposed a program to prove arithmetic circuit lower bounds through the explicit construction of elusive functions. These are polynomial maps from a low dimensional space to a high dimensional ambient space whose image is contained in no subvariety of low complexity. Here, complexity is prescribed in terms of the dimension and degree of parametric maps into the ambient space defining the subvariety. Elusive functions are abundant: finding explicit ones with parameters typical of generic polynomial maps implies Valiant's hypothesis that VP$\neq$VNP. But no such construction is known. Raz devised elusive functions with weaker parameters to derive explicit degree d polynomials in n variables requiring superlinear circuit size at depth $d=o(\log n)$. We present a new method to analyse and construct elusive functions, with coordinate maps restricted to monomials. To prove elusiveness, we identify a hitting set of points, each a tuple of roots of unity coupled based on the exponents of the monomial maps. Using Chebotarev's theorem on roots of unity, we show that for every low complexity subvariety, the function evaluated at some point in the hitting set eludes it. For this strategy to work, it suffices that the iterated sumset of a certain set of numbers (derived from the exponents) expands exponentially. We thus reduce open explicit construction problems in elusive functions to purely additive combinatorial ones, whose resolutions imply as yet unknown lower bounds. Informed by iterated sumset expansion, we devise new elusive functions. We construct explicit elusive curves of exponential degree, resolving an open problem posed by Garg, Makam, Oliveira, and Wigderson as a testament to the difficulty of elusiveness proofs. We improve Raz's superlinear bound quadratically (with circuit size to input size ratio as the metric) below $o(\log n/\log\log n)$ depths.
