Tight bounds on the degree of the representing polynomial f

Establish tight bounds on the degree of the representing polynomial f in K[u1, …, un] satisfying h = f(g1, …, gn) for h ∈ K[g1, …, gn], expressed in terms of the degree of h and the degrees of the algebraically independent generators g1, …, gn, so as to sharpen the complexity O((nL1 + n^4 + L2)·M(Δ, n)) of the Representation algorithm.

Background

The algorithm’s complexity depends critically on Δ, a degree bound for f. In special cases (e.g., pseudo-reflection groups), the paper shows deg_{W'}(f) = deg(h) and consequently deg(f) ≤ deg(h), but in general settings tight bounds are not provided.

The authors explicitly identify the lack of tight bounds on deg(f) as an open problem, noting that improved degree estimates (possibly via weighted degrees) would lead to sharper complexity bounds for their algorithm.