Seminar series
Thu, 19 May 2022
11:30 - 12:45
Thomas Scanlon
University of California, Berkeley
A $\sigma$-variety over a difference field $(K, \sigma)$ is a pair $(X, \varphi)$ consisting of an algebraic variety $X$ over $K$ and $\varphi : X \rightarrow X^{\sigma}$ is a regular map from $X$ to its transform $X^{\sigma}$ under $\sigma$. A subvariety $Y \subseteq X$ is skew-invariant if $\varphi(Y) \subseteq Y^{\sigma}$. In earlier work with Alice Medvedev we gave a procedure to describe skew-invariant varieties of $\sigma$-varieties of the form $(\mathbb{A}^n, \varphi)$ where $\varphi(x_1, \dots, x_n) = (P_1(x_1), \dots, P_n(x_n))$. The most important case, from which the others may be deduced, is that of $n=2$. In the present work we give a sharper description of the skew-invariant curves in the case where $P_2 = P_1^{\tau}$ for some other automorphism of $K$ which commutes with $\sigma$. Specifically, if $P \in K[x]$ is a polynomial of degree greater than one which is not eventually skew-conjugate to a monomial or $\pm$ Chebyshev (i.e. $P$ is "nonexceptional") then skew-invariant curves in $(\mathbb{A}^2, (P, P^{\tau}))$ are horizontal, vertical, or skew-twists: described by equations of the form $y = \alpha^{\sigma^n} \circ P^{\sigma^{n-1}} \circ \dots \circ P^{\sigma} \circ P(x)$ or $x = \beta^{\sigma^{-1}} \circ P^{\tau \sigma^{-n-2}} \circ P^{\tau \sigma^{-n-3}} \circ \dots \circ P^{\tau}(y)$ where $P = \alpha \circ \beta$ and $P^{\tau} = \alpha^{\sigma^{n+1}} \circ \beta^{\sigma^n}$ for some integer $n$. 
We use this new characterization to prove that a function $f(t)$ which satisfies $p$-Mahler equation of nonexceptional polynomial type, by which we mean $f(t^p) = P(f(t))$ for $p \in \mathbb{Q}_{+} \setminus \{1\}$ and $P \in \mathbb{C}(t)[x]$ a nonexceptional polynomial, is necessarily algebraically independent from functions satisfying $q$-Mahler equations with $q$ multiplicatively independent from $p$. 
This is a report on joint work with Khoa Dang Nguyen and Alice Medvedev available at arXiv:2203.05083.  
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