Algebraic complexity theory via the algebraic geometry and representation theory of generalised continued fractions

高端计算机可以解决极其困难的任务，但即使是最好的计算机所能做的也是有限的。这些极限是在计算复杂性理论领域中研究的。计算复杂性理论中最著名的问题是P vs NP猜想，解决这个问题的个人或团体将获得100万美元的奖金。这个问题可以用布尔电路和算法表述如下。布尔电路是对一个有限的逻辑运算序列的描述，该逻辑运算序列是对输入值列表执行的，很像计算机程序，但没有循环。给定一个布尔电路，我们想知道是否存在使其输出为TRUE的输入。P vs NP猜想表明，对于这个任务没有有效的算法。更准确地说，对于用n个符号描述的电路，回答这个问题所需的计算时间比n中的任何多项式都要快：超多项式快。这与用来回答问题的算法无关。研究人员普遍认为，我们离解决P与NP猜想还很远。许多算法的突破不是研究布尔电路，而是基于代数的洞见，例如欧几里得算法、有效的中国剩余定理、高斯消去、牛顿迭代、快速矩阵乘法、快速多项式乘法、快速傅立叶变换、椭圆曲线密码学、里德-所罗门码，或通过Gröbner基求解多项式方程组。一个更重要的例子是霍纳的方法，一种有效地计算单变量多项式的方法。1966年，Victor Pan证明，如果我们计算算术运算的次数而不是布尔运算的次数，Horner的方法是最优的。利用这种测量问题复杂性的方法，Leslie Valiant在1979年定义了P对NP问题的代数类比。在这个提议中，我们关注Valiant的VF vs VNP猜想。这一猜想表明，计算有向图中哈密顿循环的数目的算术公式所需的长度是超多项式增长的，这是组合学中的一个问题。这与P vs NP猜想密切相关：如果P等于NP，那么所谓的多项式层次结构就会崩溃，而大多数复杂性理论家认为这种情况不会发生。如果有人证明这种坍缩不会发生，那么这也意味着Karp和Lipton在1980年的结果表明VF不等于VNP，所以在某种意义上，为了取得进展，必须解决VF vs VNP猜想。我们希望在VF vs VNP猜想上取得进展，因为最近的一篇论文（Bringmann, Ikenmeyer, Zuiddam 2017）显示了算术公式大小与Horner方法的多元（即多个变量）版本之间的深刻联系。这意味着分离VF和VNP归结为在更深层次上理解多元霍纳方法。它与连分式和连续多项式有联系，最近由Charles Conley和Valentin Ovsienko（2018）研究。除了他们的工作，到目前为止，只有初步的代数、几何和算法结果是已知的（最近由Shpilka和Medini），我们的目标是提供缺失的基础。我们的目标是通过几种方式将我们的新见解与几何复杂性理论联系起来，几何复杂性理论是一种使用代数几何和表示理论的复杂性下界的数学方法，由Ketan Mulmuley和Milind Sohoni于2001年创立。这将导致证明算术公式大小下界的新方法，从而最终有助于分离VF和VNP。

High-end computers can solve extremely difficult tasks, but even the best computers have limits in what they can do. These limits are studied in the field of computational complexity theory. The most famous question in computational complexity theory is the P vs NP conjecture, where a price of $1,000,000 will be awarded to the person or group who resolves it. The question can be phrased in terms of Boolean circuits and algorithms as follows. A Boolean circuit is a description of a finite sequence of logical operations that is performed a list of input values, much like a computer program, but with no loops. Given a Boolean circuit we want to know if there exists an input that makes it output TRUE. The P vs NP conjecture states that there is no efficient algorithm for this task. More precisely, the computation time required to answer this question for a circuit that is described using n symbols grows faster than any polynomial in n: Superpolynomially fast. This is independent of the algorithm that is used to answer the question. It is generally assumed among researchers that we are very far from resolving the P vs NP conjecture though. Instead of studying Boolean circuits many algorithmic breakthroughs are based on algebraic insights, for example the Euclidean algorithm, the effective Chinese Remainder Theorem, Gaussian elimination, Newton iteration, fast matrix multiplication, fast polynomial multiplication, fast Fourier transform, Elliptic Curve Cryptography, Reed-Solomon codes, or solving systems of polynomial equations via Gröbner bases. One more important example is Horner's method, a way of evaluating polynomials in a single variable efficiently. In 1966 Victor Pan proved that Horner's method is optimal if we count the number of arithmetic operations instead of Boolean operations. Using this way of measuring the complexity of a problem, in 1979 Leslie Valiant FSR defined algebraic analogues of the P vs NP problem. In this proposal we focus on Valiant's VF vs VNP conjecture. This conjecture says that the required size of arithmetic formulas counting the number of Hamiltonian cycles in directed graphs, a problem from combinatorics, grows superpolynomially. This is closely related to the P vs NP conjecture: If P equals NP, then the so-called polynomial hierarchy collapses, which most complexity theorists think does not happen. If one proves that this collapse does not happen, then this also implies that VF is not equal to VNP by a result of Karp and Lipton in 1980, so in a sense the VF vs VNP conjecture has to be resolved in order to make progress.We hope to make progress on the VF vs VNP conjecture, because a recent paper (Bringmann, Ikenmeyer, Zuiddam 2017) shows a deep connection between arithmetic formula size and a multivariate (i.e., several variables) version of Horner's method. This means that separating VF and VNP boils down to understanding the multivariate Horner's method on a deep level. It has connections to continued fractions and the continuant polynomial, which was recently studied by Charles Conley and Valentin Ovsienko (2018). Other than their work, only preliminary algebraic, geometric, and algorithmic results are known so far (very recently by Shpilka and Medini) and we aim to provide the missing foundations. We aim to connect our new insights in several ways with Geometric Complexity Theory, a mathematical approach for complexity lower bounds that uses algebraic geometry and representation theory and that was founded in 2001 by Ketan Mulmuley and Milind Sohoni. This will lead to new ways of proving lower bounds on arithmetic formula size, which then ultimately can help to separate VF and VNP.