论文标题
代数独立性和线性差方程
Algebraic independence and linear difference equations
论文作者
论文摘要
我们考虑作用于laurent或puiseux系列的$（ϕ，σ）$的自动形态$（ϕ \ colon x \ colon x+h_1+h_1，σ\ colon x \ colon x \ colon x+apsto x+h_2）$（ x \ mapsto q_2x）$和mahler operator $（ϕ \ colon x \ mapsto x^{p_1}，\σ\ colon x \ colon x \ mapsto x^{p_2}）$。给定对线性$ ϕ $ - 平衡的解决方案$ f $，以及对线性$σ$ - 平等的解决方案$ g $，既可以先验，我们表明$ f $和$ g $在合理函数的领域上是独立的，假设相应的参数足够独立。结果，我们解决了1987年Loxton和van der Poorten提出的有关Mahler功能的猜想。我们还为代数独立性提供了$ Q $ - hypheperementic函数的应用。我们的方法提供了研究这种问题的一般策略，并基于合适的Galois理论：$σ$ -Galois线性$ ϕ $ - 方程式。
We consider pairs of automorphisms $(ϕ,σ)$ acting on fields of Laurent or Puiseux series: pairs of shift operators $(ϕ\colon x\mapsto x+h_1, σ\colon x\mapsto x+h_2)$, of $q$-difference operators $(ϕ\colon x\mapsto q_1x,\ σ\colon x\mapsto q_2x)$, and of Mahler operators $(ϕ\colon x\mapsto x^{p_1},\ σ\colon x\mapsto x^{p_2})$. Given a solution $f$ to a linear $ϕ$-equation and a solution $g$ to a linear $σ$-equation, both transcendental, we show that $f$ and $g$ are algebraically independent over the field of rational functions, assuming that the corresponding parameters are sufficiently independent. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of $q$-hypergeometric functions. Our approach provides a general strategy to study this kind of question and is based on a suitable Galois theory: the $σ$-Galois theory of linear $ϕ$-equations.
