Arithmetic Analogue of Differential Algebraic Geometry

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9730183 Buium In previous research, Buium investigated an analogue of algebraic geometry in which algebraic equations are replaced by algebraic differential equations. This geometry may be called differential algebraic geometry, and its commutative algebra counterpart is the Ritt-Kolchin differential algebra. Applications of Buium's work include the ``effective geometric Lang conjecture.'' In subsequent research, Buium developed an arithmetic analogue of differential algebraic geometry. Applications of the latter included a proof of the "effective Manin-Mumford conjecture" and an arithmetic analogue of Manin's theorem of the Kernel. The main theme of this project will be continuing this line of investigation, especially in the arithmetic direction. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.