Quantitative Methods for Systems of Polynomial-Exponential Diophantine Equations

The effective description of the set of solutions of a system of Diophantine equations over a given number field or a ring of integers arises in a wide variety of fields, from fundamental mathematics (number theory, algebraic geometry) to theoretical computer science (algorithm verification and quantum automata). Such a set is sometimes difficult to describe by simple elementary arithmetic methods but using deepest mathematical theories, one can often get compelling problems. One such example is the famous Skolems problem, which seeks an effective description of the set of zeros in a given linear recurrence sequence. This question has long been open in number theory, dating back to 1934. The main purpose of the research proposal is to provide algorithms that completely describe the set of solutions of some large classes of systems of Diophantine equations involving linear recurrence sequences, the Beal-Fermat equation with signature (p, p, l), or related Diophantine approximation problems. We plan to build these algorithms by combining existing methods such as Wiless modular approach, Schmidts subspace theorem, and Bakers method of linear forms in logarithms. As an application, we plan to develop a new cryptosystem protocol (in the field of cybersecurity) based on Diophantine analysis, since classical cryptosystems become compromised by quantum computer.