论文标题
偏斜不变的曲线和Mahler功能的代数独立性
Skew-invariant curves and the algebraic independence of Mahler functions
论文作者
论文摘要
对于$ p \ in \ mathbb {q} _+ \ smallSetMinus \ {1 \} $与一个不同的正理性数字，我们说PuisSeux系列$ f \ in \ Mathbb {c}（c}（c}（（t）） \ Mathbb {c}（t）^\ text {alg} [x] $至少两个与单一或或plus或plus或或减去chebyshev polyensial chebyshev polyenmial neke $ f（t^p）= p（f（f（t））$持有。我们表明，如果$ p $和$ q $是多重独立的，而$ f $和$ g $是$ p $ -mahler和$ q $ -mahler，则是非优先级的多项式类型的，那么$ f $和$ g $与$ \ m athbb {c}（c}（t）$。该定理被证明是由于一个更一般的定理证明，如果$ f $是$ p $ - 非远程多项式类型的mahler，$ g_1，\ ldots，g_n $，每个$ g_n $都满足与替换$ t \ t \ mapsto t^q $的差异方程，那么$ f $ f $ a $ f $与$ g_1独立于$ g_1，g_1，g_1，g_1，g_ ld。这些定理本身是$ \ mathbb {a}^2 $的分裂多项式动力学系统的偏斜曲线的精制分类的后果。
For $p \in \mathbb{Q}_+ \smallsetminus \{ 1 \}$ a positive rational number different from one, we say that the Puisseux series $f \in \mathbb{C}((t))^\text{alg}$ is $p$-Mahler of non-exceptional polynomial type if there is a polynomial $P \in \mathbb{C}(t)^\text{alg}[X]$ of degree at least two which is not conjugate to either a monomial or to plus or minus a Chebyshev polynomial for which the equation $f(t^p) = P(f(t))$ holds. We show that if $p$ and $q$ are multiplicatively independent and $f$ and $g$ are $p$-Mahler and $q$-Mahler, respectively, of non-exceptional polynomial type, then $f$ and $g$ are algebraically independent over $\mathbb{C}(t)$. This theorem is proven as a consequence of a more general theorem that if $f$ is $p$-Mahler of non-exceptional polynomial type, and $g_1, \ldots, g_n$ each satisfy some difference equation with respect to the substitution $t \mapsto t^q$, then $f$ is algebraically independent from $g_1, \ldots, g_n$. These theorems are themselves consequences of a refined classification of skew-invariant curves for split polynomial dynamical systems on $\mathbb{A}^2$.
