Optimality of the δ(n) factor in symmetric polynomial evaluation complexity

Determine whether the dependence factor δ(n) in the complexity bound δ(n)·L2 + 2 for evaluating the representing polynomial f of a symmetric polynomial h when g1, …, gn are the elementary symmetric polynomials (as in the algorithm of Gaudry–Schost–Thiéry) is optimal with respect to n; either prove optimality via lower bounds or design an algorithm with strictly smaller asymptotic dependence on n.

Background

In the discussion of prior work on evaluating representations of symmetric polynomials in terms of elementary symmetric generators, the paper cites the result of Gaudry, Schost, and Thiéry bounding the evaluation complexity by δ(n)·L2 + 2, where L2 is the cost to evaluate h and δ(n) ≤ 4^n (n!)^2.

The paper explicitly notes that those authors did not know whether the δ(n) factor is optimal. While this work addresses a more general setting by tying the cost to multivariate series multiplication, it does not resolve the optimality of δ(n) in the symmetric case, leaving a precise characterization of the best possible dependence on n as an open problem.