Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1992. / Includes bibliographical references (p. 144-145).

Thesis (Ph. D.)--University of Washington, 1986. / On t.p. "[p-adic]" appears as a Gothic "Z" with superscript "d" and subscript "p." Vita. Bibliography: leaves [66]-67.

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We study the quadratic form induced by the Bezoutian of two polynomials p and q, considering four problems. First, over R, in the separable case we count the number of configurations of real roots of p and q for which the Bezoutian has a fixed signature. Second, over Qp we develop a formula for the Hasse invariant of the Bezoutian. Third, we formulate a conjecture for the behavior of the Bezoutian in the non separable case, and offer a proof over R. We wrote a Pari code to test it over Qp and Q and found no counterexamples. Fourth, we state and prove a theorem that we hope will help prove the conjecture in the near future. / text

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, December 1999. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, August 2002. / Includes bibliographical references. Also available on the Internet.

Here we offer an introduction to the adele ring over the field of rational numbers Q and highlight some of its beautiful algebraic and topological structure. We then apply this rich structure to revisit some ancient results of number theory and place them within this modern context as well as make some new observations. We conclude by indicating how this theory enables us to extend the basic arithmetic of Q to the more subtle, complicated, and interesting setting of an arbitrary number field.

Abstract The dissertation consists of three articles in which irrationality measures for some values of certain special cases of the Gauss hypergeometric function are considered in both archimedean and non-archimedean metrics. The first presents a general result and a divisibility criterion for certain products of binomial coefficients upon which the sharpenings of the general result in special cases rely. The paper also provides an improvement concerning th e values of the logarithmic function. The second paper includes two other special cases, the first of which gives irrationality measures for some values of the arctan function, for example, and the second concerns values of the binomial function. All the results of the first two papers are effective, but no computation of the constants for explicit presentation is carried out. This task is fulfilled in the third article for logarithmic and binomial cases. The results of the latter case are applied to some Diophantine equations.

Let Zp be the ring of p-adic integers and Ep be the map x-->exp(px) where exp denotes the exponential map determined by the usual power series. It defines an exponential ring (Zp, + , . , 0, 1, Ep). The goal of the thesis is to study the model theory of this structure. In particular, we are interested by the question of the decidability of this theory. The main theorem of the thesis is: Theorem: If the p-adic Schanuel's conjecture is true, then the theory of (Zp, + , . , 0, 1, Ep) is decidable. The proof involves: 1- A result of effective model-completeness (chapters 3 and 4): If F is a family of restricted analytic functions (i.e. power series with coefficients in the valuation ring and convergent on Zp) closed under decomposition functions and such that the set of terms in the language LF= (+, . , 0, 1, f; f in F) is closed under derivation, then we prove that the theory of Zp in the language LF is model-complete. And furthermore, if each term of LF has an effective Weierstrass bound, then the model-completeness is effective. 2- A resolution of the decision problem for existential formulas (assuming Schanuel's conjecture) in chapter 5. We also consider the problem of the decidability of the structure (Op, + , . , 0, 1, |, E_p) where Op denotes the valuation ring of Cp. We give a positive answer to this question assuming the p-adic Schanuel's conjecture.