Rational functions admitting double decompositions

Abstract

Abstract. Ritt (1922) studied the structure of the set of complex polynomials with respect to composition. A polynomial P (x) is said to be indecomposable if it can be represented as P = P 1 •P 2 only if either P 1 or P 2 is a linear function. A decomposition P = P 1 •P 2 •. . .•P r is said to be maximal if all the P j are indecomposable polynomials that are not linear. Ritt proved that any two maximal decompositions of the same polynomial have the same length r, the same (unordered) set {deg(P j )} of the degrees of the composition factors, and can be connected by a finite chain of transformations each of which consists in replacing the left-hand side of the double decompositionby its right-hand side. Solutions of this functional equation are indecomposable polynomials of degree greater than 1, and Ritt listed all of them explicitly.Up until now, analogues of Ritt's theory for rational functions have only been constructed for some special classes of these functions, for instance, for Laurent polynomials (Pakovich, 2009). In this note we describe a certain class of double decompositions (1) with rational functions R j (x) of degree greater than 1. In essence, the rational functions described below were discovered by Zolotarëv as solutions of a certain optimization problem (1932). However, the double decomposition property for these functions remained little known because they had an awkward parametric representation. Below we give a representation for Zolotarëv fractions (possibly new), which resembles the well-known representation for Chebyshëv polynomials. These, by the way, are a special limit case of Zolotarëv fractions. § 1. Zolotarëv fractions and their composition property A purely imaginary parameter τ ∈ iR + defines the rectangle Π(τ ) of size 2 × |τ |:The conformal map x τ (u) taking this rectangle to the upper half-plane and fixing the three points u = ±1, 0 has the following simple form:in terms of the elliptic sine sn and the complete elliptic integral K. It is easy to deduce from the reflection principle for a conformal map that the parametric representation R(u) := x τ (u); x(u) := x nτ (u), u ∈ C, n ∈ N, defines a rational function R of degree n in the argument x depending parametrically on τ :Z n (x|τ ) := R(u(x)) = x τ • x −1 nτ .

Lower bounds for genera of fiber products

“…hold. Thus, if C has bi-degree pk, lq with k ą 1, then (12) follows from (10). On the other hand, if k " 1, then obviously B " A ˝S for some rational function S, and C is the graph x ´Spyq " 0.…”

“…The general case, however, is much less understood and known results are mostly concentrated on a study of either decompositions of special types of functions or functional equations of a special form (see e.g. [6], [10], [16], [23], [39], [40], [42], [46], [51], [55]). Notice also that, by the Picard theorem, any algebraic curve that can be parametrized by functions meromorphic on C has genus zero or one.…”

“…Let A and B rational functions of degree n and m correspondingly and C an irreducible component of the curve E A,B of bi-degree pk, lq such that k ą 1. Then (10) gpEq ą 2 ´k `m npn ´1q . .…”

Recent Progress in Optimization of Multiband Electrical Filters

“…Alternation points different from critical points of the fraction correspond to four corners of the large rectangle. Zeros/poles of the fraction correspond to u = lτ with even/odd l. Remark 1 Zolotarëv fractions share many interesting properties with Chebyshëv polynomials as the latter are the special limit case of the former [18,19]. For instance, the superposition of suitably chosen Zolotarëv fractions is again a Zolotarëv fraction.…”