Generalized Integral Bases and Applications in Creative Telescoping

The notion of integrality known for algebraic numbers and algebraic functions has been extended to D-finite functions by Kauers and Koutschan in 2015. One aim of the present thesis is to further extend this notion to the case of P-recursive sequences. In order to do so, we formulate a general algorithm for finding all integral elements of valued vector spaces and then show that this algorithm includes not only the algebraic and D-finite cases but also covers the case of P-recursive sequences. A second aim of the thesis is to explore the applications of integral bases in symbolic integration via creative telescoping. Bronstein's lazy Hermite reduction is a symbolic integration technique that reduces algebraic functions to integrands with only simple poles without the prior computation of an integral basis. We sharpen the lazy Hermite reduction by combining it with the polynomial reduction to solve the decomposition problem of algebraic functions. The sharpened reduction is then used to design a reduction-based telescoping algorithm for algebraic functions in two variables. Trager's Hermite reduction solved the integration problem for algebraic functions via integral bases, which was extended to Fuchsian D-finite functions. We remove the Fuchsian restriction and present Hermite reduction for general D-finite functions. It reduces the pole orders of integrands at finite places, but may not reduce to simple poles as in the algebraic and Fuchsian D-finite cases. Instead of using the polynomial reduction, we develop Hermite reduction at infinity to reduce the pole order of D-finite functions at infinity via local integral bases at infinity. Combining Hermite reduction at finite places and at infinity, we obtain a new reduction-based telescoping algorithm for D-finite functions in two variables.