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Preprint Number 2144

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2144. Alice Medvedev, Khoa Dang Nguyen, and Thomas Scanlon
Skew-invariant curves and the algebraic independence of Mahler functions

Submission date: 9 March 2022


For p in ℚ_+ \{1} a positive rational number different from one, we say that the Puisseux series f in ℂ((t))^{alg} is p-Mahler of non-exceptional polynomial type if there is a polynomial P in ℂ(t)^{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 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, ... , g_n each satisfy some difference equation with respect to the substitution t ↦ t^q, then f is algebraically independent from g_1, ... , g_n. These theorems are themselves consequences of a refined classification of skew-invariant curves for split polynomial dynamical systems on 𝔸^2.

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Full text arXiv 2203.05083: pdf, ps.

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