The research contemplated in this proposal is directed at two related Diophantine problems about polynomials. The first problem is to better understand the distribution of values of the local and global Mahler measure of polynomials with coefficients restricted to either algebraic number fields or their completions. The main technical tool used to accomplish this is to study the Mellin transform of the distribution function associated to Mahler's measure. Then Mellin inversion techniques from analysis are used to obtain estimates or explicit formulas for the distribution functions. A further aspect of the research is to interpret the results in terms of the heights of algebraic numbers. The second problem is to consider the Wronskian of a collection of linearly independent polynomials with coefficients in an algebraic number field or the completion of such a field. We seek an inequality which relates the local and global Mahler measures of the Wronskian with the local and global Mahler measures of the original set of linearly independent polynomials. This is well understood for two polynomials, but only approximate results are known for a collection of three or more polynomials. Again there is an interpretation in terms of the heights of the roots of the polynomials, and this turns out to be related to the ABC-inequality of Stothers and Mason. A further aspect of this research is to work out applications to certain Diophantine problems.
A Diophantine equation is an equation to be solved in integers. The study of such equations is one of the oldest parts of number theory and continues to attract some of the best mathematical researchers. The tools developed in modern mathematics to study Diophantine equations can be extremely technical, and often involve related issues that can be described under the general title of "Diophantine problems". For example, one might study inequalities to be solved in integers, or to be solved in polynomials with integer coefficients. The research considered in this proposal is directed at exactly such Diophantine inequalities for polynomials. In all fields of science there is generally a great intellectual distance between the highly technical efforts of pure researchers and the eventual applications to business, technology or engineering. We do know that the area of Diophantine problems has important applications to computer problems, to the construction of encryption systems, and to other technological questions. A solution to the problems contemplated in this proposal would further our understanding of pure mathematical questions which belong to this important field. Also, this proposal will fund the research of several graduate students at the University of Texas. These students will benefit from the opportunity to concentrate on the research efforts, to travel and talk about their work at scientific meetings, and to consult with other researchers working on related problems. A further important goal of this work is the completion of a monograph written by the PI with the title ``Heights, Mahler measure and Diophantine inequalities". Over a period of many years the PI has written notes for graduate students in classes on this topic that the PI has taught at the University of Texas. These notes will be collected, edited and somewhat expanded into a monograph.
