Composite rational functions expressible with few terms. (English) Zbl 1244.12001
Let \(k\) be a field of characteristic zero, and let \(f(x)=P(x)/Q(x)\in k(x)\) be a rational function which is “lacunary”, in the sense that the polynomials \(P\) and \(Q\) have altogether at most a given number \(\ell\) of terms (the polynomials \(P\) and \(Q\) need not be coprime). The paper studies the possible non-trivial decompositions of such a function, namely, the expressions of type \(f(x)=g(h(x))\) where \(g\) and \(h\) are rational functions of degree \(>1\). Let us call a decomposition of this type exceptional if \(h(x)=\lambda(ax^n+bx^{-n})\) for \(a,b\in k\), \(n\in{\mathbb N}\) and \(\lambda\in\text{PGL}_2(k)\).
The main theorem states the following: if \(f(x)=g(h(x))\) and the decomposition is neither trivial nor exceptional, then \(\deg g \leq 2016 \cdot 5^\ell\).
The main theorem states the following: if \(f(x)=g(h(x))\) and the decomposition is neither trivial nor exceptional, then \(\deg g \leq 2016 \cdot 5^\ell\).
Reviewer: Roberto Dvornicich (Pisa)
MSC:
| 12E05 | Polynomials in general fields (irreducibility, etc.) |
| 11R09 | Polynomials (irreducibility, etc.) |
| 37P05 | Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps |
References:
| [1] | Avanzi, R., Zannier, U.: The equation f (X) = f (Y ) in rational functions X = X(t ), Y = Y (t ). Compos. Math. 139, 263-295 (2003) · Zbl 1050.14020 · doi:10.1023/B:COMP.0000018136.23898.65 |
| [2] | Beukers, F., Smyth, C. J.: Cyclotomic points on curves. In: Number Theory for the Mil- lennium, I (Urbana, IL, 2000), A K Peters, Natick, MA, 67-85 (2002) · Zbl 1029.11009 |
| [3] | Bilu, Y., Tichy, R.: The Diophantine equation f (x) = g(y). Acta Arith. 95, 261-288 (2000) · Zbl 0958.11049 |
| [4] | Brownawell, D., Masser, D.: Vanishing sums in function fields. Math. Proc. Cambridge Philos. Soc. 100, 427-434 (1986) · Zbl 0612.10010 · doi:10.1017/S0305004100066184 |
| [5] | Dixon, J. D., Mortimer, B.: Permutation Groups. Grad. Texts in Math. 163, Springer, New York (1996) · Zbl 0951.20001 |
| [6] | Dvornicich, R., Zannier, U.: Cyclotomic Diophantine problems (Hilbert irreducibility and invariant sets for polynomial maps). Duke Math. J. 139, 527-554 (2007) · Zbl 1127.11040 · doi:10.1215/S0012-7094-07-13934-6 |
| [7] | Fried, M.: On a conjecture of Schur. Michigan Math. J. 17, 41-55 (1970) · Zbl 0169.37702 · doi:10.1307/mmj/1029000374 |
| [8] | Fried, M.: Arithmetical properties of function fields. II. The generalized Schur problem. Acta Arith. 25, 225-258 (1974) · Zbl 0229.12020 |
| [9] | Gabrielov, A., Vorobjov, N.: Complexity of computations with Pfaffian and Noetherian func- tions. In: Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, NATO Sci. Ser. II Math. Phys. Chem. 137, Kluwer, Dordrecht, 211-250 (2004) |
| [10] | Lang, S.: Introduction to Algebraic and Abelian Functions. 2nd ed., Grad. Texts in Math. 89, Springer, New York (1982) · Zbl 0513.14024 |
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