Diophantine Approximation and Arithmetic of Polynomials

University Of Colorado At Boulder, Boulder CO

Investigators

Abstract

The Proposer intends to work on a number of diophantine questions, with particular emphasis on mixed polynomial-exponential type equations in several variables, and on Diophantine approximation in positive characteristic. For polynomial-exponential equations, the goal will be to obtain bounds for the number of solutions, which are independent of the coefficients of the equations. As for approximation in fields of power series of positive characteristic, the aim will be to exhibit new classes of algebraic series with explicit continued fraction expansions, and to obtain new information on "Roth exponents". The Proposer further plans to address the question (connected with a former conjecture of Artin) of how many variables are needed to guarantee that a form of degree d with p-adic coefficients has a nontrivial zero, and he is interested in a certain question on polynomials with applications to Sidon sequences. With the advent of computers, and in modern cryptography, ``discrete'' mathematical questions, in particular questions involving integers, have turned out to be of increasing importance. The Proposer intends to continue his work on "Diophantine equations", i.e., to study integer solutions of equations. Whereas most work up to now has been on polynomial equations, the new project will also involve exponential equations, involving terms with ``exponentially fast'' growth. There will be applications to linear recurrence sequences, of which the best known is the Fibonacci sequence, which occurs, e.g., as data relating to some plants. The approximation of more complicated numbers by the more simple "rational numbers" (i.e., ordinary fractions) is at the forefront of much current work. For instance, how well can the number "pi" be approximated? The Proposer intends to work on the analogous question, on how well a more general function can be approximated by the simpler rational functions. Again of particular interest here is the discrete (positive characteristic) case.