论文标题
Mahler离散残基和理性功能的总结性
Mahler discrete residues and summability for rational functions
论文作者
论文摘要
我们为有理功能构建Mahler离散残基，并表明它们完全阻碍了Mahler总结性问题，即确定给定的有理函数$ f（x）$是$ g（x^p）-g（x）-g（x）$的某些合理函数$ g（x）$和integer $ p> p> 1 $。这扩展到了Mahler案例，由Chen and Singer开发的类似的概念，属性和应用（在移位案例中）和$ Q $ discrete残留物（在$ Q $ - 差异案例中）。在途中，我们定义了一些其他概念，这些概念有望可以解决涉及理性功能的Mahler差异领域的相关问题，包括特别是在Mahler差异方程式的（差异）Galois理论中的伸缩问题和问题。
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.
