Alexey I. Ovchinnikov proposes to develop efficient algorithms that determine the differential, difference, and algebraic structure of solutions of difference equations using methods of differential, difference, and computer algebra. The investigator has successfully contributed to the development of computational differential ideal theory and the Galois theory of linear differential equations. He will apply these results and methods to give efficient algorithms that compute properties of systems of difference equations. These properties are reflected in characteristic sets and Galois groups of the equations. In particular, the investigator will develop algorithms to reduce systems of non-linear difference equations to simpler systems. These algorithms test consistency of the input system and eliminate variables from the equations of the system. He will also give a Galois theory of linear difference equations with difference parameters to study difference algebraic dependences among solutions of difference equations.
