The Laurent Phenomenon and Discrete Integrable Systems
The Laurent Phenomenon and Discrete Integrable Systems
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发表时间：
2016
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通讯作者：
Mase
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作者：
Mase
罗朗现象是离散方程初值问题的解表示为初值的罗朗多项式的性质。这一概念源于对簇代数的研究，已知任何簇变量都是初始簇变量的洛朗多项式。在这篇文章中，我们把与簇代数的联系放在一边，研究劳伦特现象本身。我们将解释在可积系统领域中出现的大多数离散双线性方程都表现出这种现象，并且我们将讨论它与可积性的关系。最后，我们将介绍一种计算依赖于这种现象的代数熵的技术。为简洁起见，我们将省略我们所提出的定理的大多数证明。
The Laurent phenomenon is the property that the solution to an initial value problem of a discrete equation is expressed as a Laurent polynomial of the initial values. This concept has arisen from the study of cluster algebras, for which it is known that any cluster variable is a Laurent polynomial of the initial cluster variables. In this paper, we leave the connection with cluster algebras aside and study the Laurent phenomenon for its own sake. We will explain that most of the discrete bilinear equations that appear in the field of integrable systems exhibit this phenomenon and we shall discuss its relation to integrability. Finally, we shall introduce a technique for calculating the algebraic entropies relying on this phenomenon. For reasons of brevity we shall omit most proofs of the theorems we present.