Asymptotic complexity of iterated Hermite reduction (HermiteList)

Establish whether the theoretical worst-case complexity of Algorithm 1 (HermiteList), which iteratively applies classical Hermite reduction to a proper rational function in K(x) until only simple poles remain, is asymptotically the same as the complexity of performing a single Hermite reduction, as a function of the degree of the denominator polynomial.

Background

The paper proposes Algorithm 1 (HermiteList) to iteratively apply classical Hermite reduction to a proper rational function f ∈ K(x), producing a sequence of rational functions whose denominators become squarefree and ultimately isolate simple poles. This enables subsequent algorithms to compute discrete residues and reduced forms efficiently without full factorization.

Although HermiteList applies Hermite reduction multiple times—up to the highest pole order—the authors expect that successive inputs shrink quickly, so the first step essentially dominates. Formalizing this expectation requires a rigorous worst-case complexity analysis showing that the iterated approach does not asymptotically exceed the cost of a single Hermite reduction when measured in terms of the denominator’s degree.