A note on sparse polynomial interpolation in Dickson polynomial basis

Abstract

The sparsity t≪ deg(f) with respect to the basis Pn has been exploited—since [9] —in interpolation algorithms that reconstruct the degree/coefficient expansion (δj, cj)1≤j≤t from values ai = f(γi) at the arguments x ← γi ∈ K. Current algorithms for standard and Chebyshev bases use i = 1, . . . , N = t + B values when an upper bound B ≥ t is provided on input. The sparsity t can also be computed “on-the-fly” from N = 2t+ 1 values by a randomized algorithm which fails with probability O(ǫ deg(f)), where ǫ≪ 1 can be chosen on input. See [3] for a list of references. This note considers Dickson Polynomials for the basis in which a sparse representation is sought. Wang and Yucas [10, Remark 2.5] define the n-th degree Dickson Polynomials Dn,k(x, a) ∈ K[x] of the (k + 1)’st kind for a parameter a ∈ K, a 6= 0, and k ∈ Z≥0, k 6= 2 recursively as as follows: