报告摘要：
| D-finite functions are solutions of systems of linear partial differential equations with polynomial coefficients of special type. This class of functions has been systematically investigated by R. Stanley in his book Enumerative Combinatorics (Volume II) as basic generating functions in combinatorics. In this talk, we will present some results related to the local and global studies of D-finite functions. Singularity analysis explores the local behavior of analytic functions. In the local aspect, we will study the apparent singularities of D-finite functions and the corresponding desingularization algorithms. In the global aspect, we prove that a multivariate D-finite power series with coefficients from a finite set is rational. This generalizes a rationality theorem of van der Poorten and Shparlinski in 1996. As an application, we will show how this result can be used to study the nonnegative integer points on algebraic varieties. This talk is based on recent joint works with Jason P. Bell, Manuel Kauers, Ziming Li, Michael Singer and Yi Zhang.
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