## Mahler Functions and TranscendenceThis book is the first comprehensive treatise of the transcendence theory of Mahler functions and their values. Recently the theory has seen profound development and has found a diversity of applications. The book assumes a background in elementary field theory, p-adic field, algebraic function field of one variable and rudiments of ring theory. The book is intended for both graduate students and researchers who are interested in transcendence theory. It will lay the foundations of the theory of Mahler functions and provide a source of further research. |

### Contents

Transcendence theory of Mahler functions of one variable | 1 |

Transcendence theory of Mahler functions of several variables | 33 |

Algebraic independence of Mahler functions and their values | 78 |

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a₁ algebraic integers algebraic number field algebraic over C(z algebraically independent assume assumption c₁ characteristic root Comparing the coefficients complete the proof complex numbers contradiction converges coprime coprime polynomials define deg b(z deg Q degree denote differential field differential field extension eigenvalues fi(z field extension finite fm(z formal power series functional equation h₁ Hence homogeneous polynomial independent over Q infinitely irreducible Kumiko Nishioka Let f(z linear recurrences linearly independent log H log H(I Mahler functions Math matrix modulo nonnegative integers nonzero algebraic numbers nonzero polynomial obtain p₁ polynomial Q positive integer prime ideal Proposition prove root of unity satisfies the functional sequence subset sufficiently large Suppose Theorem 1.3 transcendence transcendence theory transcendental numbers unmixed homogeneous ideal variables vector Σ Σ