Gessel象限行走的一个初等解
An elementary solution of Gessel's walks in the quadrant
AI总结：
本文针对Gessel象限行走这一长期缺乏初等解的D-有限模型，提出一个初等且构造性的证明，并同时解决其他若干经计算机代数证明为代数的多步象限模型。
摘要：
大约在2000年，Ira Gessel猜想：在象限N^2中、从原点(0,0)出发并回到原点(0,0)、且步集为{E,NE,W,SW}的格路径数目具有简单的超几何形式。在随后的十年里，该问题被重新纳入对限制在象限内的小步行走（即步集属于{-1,0,1}^2的行走）的系统研究中。这类行走的生成函数是偏离散微分方程的典型解。目前，已经可以根据生成函数的性质（代数的、D-有限的或非D-有限的）对象限行走进行完整分类，但Gessel行走仍然令人困惑，因为它们是23个D-有限模型中唯一尚未获得初等解的模型。相反，Gessel猜想最初是在2008年通过一种富有创造性的计算机代数方法得到证明的。一年后，相关的三变量生成函数被一项计算机代数上的杰作证明是代数的。最近，这一结果又通过复杂的复分析工具被重新证明。本文给出一个初等且构造性的证明。我们的方法还解决了最近通过计算机代数被证明是代数的其他象限模型（带有多重步集）。
英文摘要：
Around 2000, Ira Gessel conjectured that the number of lattice walks in the quadrant N^2, starting and ending at the origin (0,0) and taking their steps in {E,NE,W,SW} had a simple hypergeometric form. In the following decade, this problem was recast in the systematic study of walks with small steps (that is,steps in {-1,0,1}^2) confined to the quadrant. The generating functions of such walks are archetypal solutions of partial discrete differential equations.A complete classification of quadrant walks according to the nature of their generating function(algebraic, D-finite or not) is now available, but Gessel'swalks remained mysterious because they were the only model among the 23D-finite ones that had not been given an elementarysolution. Instead, Gessel's conjecture was first proved usingan inventive computer algebra approach in 2008. A year later, the associated three-variate generating function was proved to be algebraic by a computer algebra tour de force. This was re-proved recently using elaborate complex analysis machinery. We give here an elementary and constructive proof. Our approach also solves other quadrant models (with multiple steps) recently proved to be algebraic via computer algebra.
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