6. Modular Hermite Integration

Abstract

The problem of rational function integration that we discuss in this chapter is, given two nonzero polynomials $f, g \in \mathbb{Z}[x], to compute \int (f/g)$. Most undergraduate calculus textbooks contain a solution by factoring the denominator g into linear polynomials over the complex numbers (or at most quadratic polynomials over the real numbers) and performing a partial fraction decomposition. For rational functions with only simple poles, this algorithmfirst appears in Johann Bernoulli (1703). For symbolic computation, this approach is inefficient since it involves polynomial factorization and computation with algebraic numbers, and the algorithms implemented in most computer algebra systems pursue a different approach due to Hermite (1872).