On the integration of elementary functions: computing the by Miller B.L.

By Miller B.L.

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A Gr¨obner Basis for Qi using lexicographic order z > t is {vi , z − H(t)}. 44 in [1]. For clarity, we state the theorem here: Let I be an ideal of the ring K[X1 , . . , Xn ] and assume that I has a Gr¨obner Basis, G, such that G has n elements and LT (gi ) = Xiν1 νi ≥ 1 for 1 ≤ i ≤ n using lexicographic order Xn > Xn−1 > · · · > X1 . Assume further that for 1 ≤ i ≤ n there does not exist a representation gi = f1 f2 + i−1 j=1 qj gj with f1 , f2 , q1 , . . , qi−1 ∈ K[t, z] such that f1 , f2 = 0 and degXi (f1 ) < degXi (gi ) and degXi (f2 ) < degXi (gi ).

15] Hungerford, Thomas W. Algebra. Springer, 1974. [16] Kauers, Manuel. Integration of algebraic functions: A simple heuristic for finding the logarithmic part. ACM, 2008. 41 Texas Tech University, Brian L. Miller, May 2012 [17] Knapp, Anthony W. Advanced Algebra. Birk¨auser, 2007. [18] Kuntz, Ernst. Introduction to Commutative Algebra and Algebraic Geometry. Birk¨auser, 1985. [19] Lang, Serge. Abelian Varieties. Springer, 1959. , and R. Rioboo. Integration of rational functions: Rational computation of the logarithmic part.

7] Cox, David, John Little, and Donal O’Shea. Springer, second edition, 2005. Using Algebraic Geometry. [8] Cox, David, John Little, and Donal O’Shea. Ideals, Varieties, and Algorithms. Springer, third edition, 2007. , and Z. Hajto. Introduction to Differential Galois Theory. Politechnika Krakowska, 2007. [10] Czichowski, G¨ unter. A note on gr¨obner bases and integration of rational functions. Journal of Symbolic Computation, 20(2):163–167, 1995. [11] James H. Davenport. On the Integration of Algebraic Functions.

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