Explicit algebraic solution of Zolotarev’s First Problem for low-degree polynomials, Part II

With recourse to [41], we consider three algorithms for explicitly solving, by algebraic means, Zolotarev’s First Problem (ZFP) of 1868 which is described e.g. in [1, 5, 26, 27]. We avoid the application of elliptic functions by drawing first on three tentative forms Zn,s,α,β (1 < α < β) of the sought-for monic proper Zolotarev polynomial Zn,s (n ≥ 4,s > tan2(π/(2n))). In order to compute then the compatible α = α0 and β = β0, so that Zn0,s0,α0,β0 = Zn0,s0 will hold for a prescribed degree n = n0 and prescribed intrinsic parameter s = s0, we draw on three intertwined variants and deploy them exemplarily to the third tentative form (not considered in [41]). We conclude that our first tentative form constitutes, in conjunction with our third variant, a deterministic algebraic algorithm for solving ZFP, which is advantageous with respect to complexity reduction. Three related algebraic algorithms from literature for solving ZFP, [20, 28, 48], are examined, refined and exemplified. Further existing non-elliptic approaches to ZFP, including the one by means of parametrization of algebraic curves [44], are referenced and annotated. Explicit representations of Zn,s in the algebraic power form (unexampled if n > 7) and novel characteristics, which facilitate the algebraic construction of Zn,s, are provided and additionally stored, for n ≤ 13, in an online ZFP-repository.