Paperfolding, Automata, and Rational Functions - Diagonals and ...

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As said, in characteristic zero, neither diagonals, nor Hadamard products, preserve algebraicity. For the latter, the standard example is (1 − 4X) −1/2 = X “ 2h ” X h h ; but X “ 2h h is not algebraic. The first remark is just the useful identity “ 2h ” h = (−4) h“ −1/2 ” h ”2 X h Facts such as this are of interest to logicians.Then, introductory calculus shows the latter series is given by the integral Z π/2 2 π 0 dt q (1 − 16X sin 2 t) This is a complete elliptic integral well known not to represent an algebraic function. . 29
A Beautiful Transcendence Argument For general odd k the question of the transcendence of the Hadamard powers Fk(X) = X “ ”k 2h X h h had long been open, until delightfully settled by a remark of Sharif and Woodcock. On the one hand, one views Fk as defined over Fp , so that certainly F p k = Fk and plainly Fk has degree dividing p − 1. However, it is easy to see that given any lower bound r there are infinitely many odd primes p so that p − 1 is divisible only by 1, 2, and primes greater than r . Exercise. One also confirms fairly readily that Fk is neither rational nor of degree 2 over Fp for infinitely many p. On the other hand, if Fk defined over Q were algebraic, say of degree r , then obviously its reduction mod p also is algebraic of degree at most r . 30
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Paperfolding, Automata, and Rationa
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Paperfolding, Automata, and Rationa
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Paperfolding Take a rectangular she
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A Mahler Functional Equation Aside.
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Next, if we pair the sequence Regul
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Next, if we pair the sequence Regul
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The uniform, or regular, 2-substitu
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Characteristic Functions I found th
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An Algebraic Equation in Characteri
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These remarks show that the paperfo
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These remarks show that the paperfo
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The Thue-Morse Sequence 0 1 10 11 1
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The Thue-Morse Sequence 0 1 10 11 1
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Euler’s Identity and a Functional
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An Algebraic Equation The function
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A Counter-example in Analysis I cla
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The Shapiro Sequence Consider, the
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A Remark on the Shapiro Function Se
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Transcendence of Automatic Numbers
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Page 122 and 123: Complexity of a Sequence Given an i
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Page 127 and 128: Algebraicity and Automaticity The p
Page 135 and 136: Comments on the Proof The best proo
Page 141 and 142: But for expansions over the complex
Page 143 and 144: But for expansions over the complex
Page 145: But for expansions over the complex
Page 148 and 149: Power Series in Several Variables R
Page 158 and 159: Diagonals and Hadamard Products The
Page 160 and 161: Diagonals and Hadamard Products The
Page 162 and 163: A Theorem of Furstenberg It follows
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Page 170 and 171: As said, in characteristic zero, ne
Page 174 and 175: A Beautiful Transcendence Argument
Page 176 and 177: A Beautiful Transcendence Argument
Page 178 and 179: Breaking Up in Characteristic p The
Page 184 and 185: Algebraic power series in character
Page 190 and 191: Hence, since Fp is finite, there ar
Page 192 and 193: Hence, since Fp is finite, there ar
Page 194 and 195: Suppose the series P aνx ν is gen
Page 202 and 203: So we have: Theorem. P aνx ν ∈
Page 210 and 211: The Lifting Theorem Looking careful
Page 216 and 217: Diagonals of Rational Functions Now
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Diagonals of Rational Functions Now
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Diagonals of Rational Functions Now
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Whilst (i) and (ii) are reasonably
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References MICHEL DEKKING, MICHEL M
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