On composite rational functions (Q2874929)

This article deals with composite rational functions, whose total number of zeros and poles is bounded. The number of zeros and poles is fixed, while the actual values of the zeros and poles, and their multiplicities, are considered as variables. They may be viewed as \textit{lacunary rational functions}, in analogy with lacunary polynomials, which have been extensively studied.NEWLINENEWLINE The work presented is based on the important result of \textit{C. Fuchs} and the first author [Proc. Am. Math. Soc. 139, No. 1, 31--38 (2011; Zbl 1234.11156)] who give, for every positive integer \(n\), a complete description of all rational functions \(f,g\) and \(h\) over an algebraically closed field \(k\) of characteristic zero such that \(f(x)=g(h(x))\) with \(f\) having at most \(n\) zeros and poles altogether. The proof contains an algorithm which, for a given bound \(n\), provides all the data for \(f,g\) and \(h\). The description is given in terms of some integer \(J\) and equations defining affine algebraic varieties \({\mathcal V}_i\subset\mathbb A^{n+t_i}\) over \(\mathbb Q\) effectively computable only in terms of \(n\). (The \(t_i\) are determined by partitions of \(\{1,\ldots,n\}\).)NEWLINENEWLINEIn the article under review, the authors give the results of computations done using a MAGMA implementation of the Fuchs-Pethő algorithm, described in Section 3. They compute the appropriate varieties and also provide parametrizations of the possible solutions. In Section 4 they prove Theorem 1: If \(f\) has \(3\) zeros and poles altogether and \(g\) is not of the shape \((\lambda(x))^m, \lambda\in \mathrm{PGL}_2(k), m\in\mathbb N\), then \(f\) is equivalent to eitherNEWLINENEWLINE(a) \(\frac{(x-\alpha_1)^{k_1}(x+\frac{1}{4}-\alpha_1)^{2k_2}}{(x-\frac{1}{4}-\alpha_1)^{2k_1+2k_2}}\) for some \(\alpha_1\in k\) and \(k_1,k_2\in\mathbb Z, k_1+k_2\neq 0\) orNEWLINENEWLINE(b) \(\frac{(x-\alpha_1)^{2k_1}(x+\alpha_1-2\alpha_2)^{2k_2}}{(x-\alpha_2)^{2k_1+2k_2}}\) for some \(\alpha_1,\alpha_2\in k\) and \(k_1,k_2\in\mathbb Z, k_1+k_2\neq 0\).NEWLINENEWLINE In Sections 5 and 6 the authors show examples of computations for \(n=4,5,6\) and \(7\). They mention that they have computed all the varieties for \(n=5\), and that their last example corresponds to one of \textit{M. Ayad} and \textit{P. Fleischmann} [J. Symb. Comput. 43, No. 4, 259--274 (2008; Zbl 1132.12304)], viz. \(f=\frac{x^4-8x}{x^3+1}, g=\frac{x^2+4x}{x+1}, h=\frac{x^2-2x}{x+1}\).NEWLINENEWLINEFor the entire collection see [Zbl 1279.00053].