Graduate Thesis Or Dissertation
 

Mahler's order functions and algebraic approximation of p-adic numbers

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/wp988n203

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  • If P is an integer polynomial denote the degree of P by ∂(P) and let H(P) be the maximum of the absolute value of the coefficients of P. Define Λ(P)=2[superscript ∂(P)]H(P) and for a fixed prime p let C[subscript p] denote the completion of the algebraic closure of the p-adic numbers. We generalize the order function of Mahler to the p-adic numbers by associating each θ∈C[subscript p] to the function O(u|θ) = max log (1/|P(θ)|), where |*| denotes the p-adic absolute value and the maximum is taken over all integer polynomials P(x) ∈ Z[x] satisfying Λ(P) ≤ u and P(θ) ≠ 0. Similarly, define O*(u|θ) = max log (1/|θ-α|) where the maximum is now taken over the set of all algebraic numbers α ≠ θ with minimal polynomial m satisfying Λ(m) ≤ u. Placing a partial order >> and equivalence relation ≍ on both O and O* induces two corresponding partial orders and equivalence relations on Cp. In this thesis we demonstrate several results concerning O and O*. In particular, it is proved that if θ is algebraic over Qp then θ must satisfy O*(u|θ) >> log u and if θ,η ∈ Cp that are algebraically dependent over the Q then O(u|θ) ≍ O(u|η). Also, under certain conditions, given a function g we construct θ ∈ Cp such that g(c'(log log u)[superscript ½) << O*(u|θ) and O*(u|θ) << g(c log log u), where c and c' are positive constants. The transcendence type considers the limiting behavior of O and O* and will be used to prove that under certain conditions O and O* behave similarly. Finally, given τ ≥ (3+√5)/2 we demonstrate that it is possible to construct elements of Cp with transcendence type equal to τ.
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