Abstract
We introduce in this chapter the order function at an element, which will be our main tool later when we prove the correctness of the integration algorithm. The usefulness of this function is that it maps elements of arbitrary unique factorization domains into integers, so applying it on both sides of an equation produces equations and inequalities involving integers, making it possible to either prove that the original equation cannot have a solution, or to compute estimates for the orders of its solutions. Therefore it is used in many contexts besides integration, for example in algorithms for solving differential equations. While we use only the order function at a polynomial in the integration algorithm, we introduce it here in unique factorization domains of arbitrary characteristic, and study its properties in the general case when the order is taken at an element that is not necessarily irreducible.
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