Mahler Discrete Residues and Summability for Rational Functions
Mahler Discrete Residues and Summability for Rational Functions
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DOI：
10.1145/3476446.3536186
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发表时间：
2022-02
期刊：
影响因子：
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通讯作者：
Carlos E. Arreche;Yi Zhang
中科院分区：
文献类型：
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作者：
Carlos E. Arreche;Yi Zhang
我们构造了有理函数的Mahler离散留数，并证明了它们构成了Mahler可和性问题的一个完全障碍，该问题是确定给定的有理函数f（x）对于某个有理函数g（x）$和一个整数p > 1$是否具有g（x^p）-g（x）$的形式.这扩展到马勒的情况下类似的概念，性质，和应用程序的离散残留（在移位的情况下）和q-离散残留（在q-差的情况下）开发的陈和辛格。沿着的方式，我们定义了几个额外的概念，承诺是有用的解决相关问题，涉及马勒差分领域的合理功能，特别是包括伸缩问题和问题（微分）伽罗瓦理论的马勒差分方程。
We construct Mahler discrete residues for rational functions and show that they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $g(x^p)-g(x)$ for some rational function $g(x)$ and an integer $p > 1$. This extends to the Mahler case the analogous notions, properties, and applications of discrete residues (in the shift case) and q-discrete residues (in the q-difference case) developed by Chen and Singer. Along the way we define several additional notions that promise to be useful for addressing related questions involving Mahler difference fields of rational functions, including in particular telescoping problems and problems in the (differential) Galois theory of Mahler difference equations.