A note on Gröbner bases and integration of rational functions
Given a rational function \(A(x)/D(x)\), let \(R(z)\) denote the resultant of \(A(x) - zD' (x)\) and \(D(x)\), regarded as polynomials in \(x\). Using an algorithm due to \textit{B. M. Trager} [in: Symbolic and algebraic computation, Proc. 1976 ACM Symp., Yorktown Heights/NY, 219-226 (1976; Zbl 0498.12005); also see \textit{D. Lazard} and \textit{R. Rioboo}, J. Symb. Comput. 9, No. 2, 113-115 (1990; Zbl 0723.68053)], the logarithmic part of the integral of \(A(x)/D(z)\) can be found by using the remainders that occur in the calculation of \(R(z)\) by the subresultant algorithm. The author presents an alternative way to obtain the logarithmic part by using a Gröbner basis. If \(I\) is the ideal generated by \(A(x) - zD' (x)\) and \(D(x)\), then the author shows that one can easily find the logarithmic part of the integral from the reduced Gröbner basis for \(I\) with respect to the lexicographic order with \(x > z\). The proof uses the fact that \(I\) is a zero-dimensional ideal whose zeros have distinct \(x\)-coordinates. No claim is made that this method is superior, from a computational standpoint, to Trager's algorithm.
scientific article; zbMATH DE number 871128