Let be a -th order linear recurrence sequence having polynomial characteristic roots, one of which has degree strictly greater than the others. Moreover, let be a given integer. We ask for such that the equation is satisfied for a polynomial with and some polynomial with . We prove that for all but finitely many these decompositions can be described in “finite terms” coming from a generic decomposition parameterized by an algebraic variety. All data in this description will be shown to be effectively computable.
@article{AMBP_2019__26_1_1_0, author = {Clemens Fuchs and Christina Karolus}, title = {Composite values of polynomial power sums}, journal = {Annales Math\'ematiques Blaise Pascal}, pages = {1--24}, publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal}, volume = {26}, number = {1}, year = {2019}, doi = {10.5802/ambp.380}, language = {en}, url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.380/} }
Clemens Fuchs; Christina Karolus. Composite values of polynomial power sums. Annales Mathématiques Blaise Pascal, Tome 26 (2019) no. 1, pp. 1-24. doi : 10.5802/ambp.380. https://ambp.centre-mersenne.org/articles/10.5802/ambp.380/
[1] The equation in rational functions , , Compos. Math., Volume 139 (2003) no. 3, pp. 263-295 | Article | MR 2041613 | Zbl 1050.14020
[2] Combinatorial Diophantine equations and a refinement of a theorem on separated variables equations, Publ. Math., Volume 82 (2013) no. 1, pp. 219-254 | MR 3034377 | Zbl 1289.11035
[3] The Diophantine equation , Acta Arith., Volume 95 (2000) no. 3, pp. 261-288 | Article | Zbl 0958.11049
[4] Decomposition of polynomials and approximate roots, Proc. Am. Math. Soc., Volume 138 (2010) no. 6, pp. 1989-1994 | Article | MR 2596034 | Zbl 1193.13008
[5] Vanishing sums in function fields, Math. Proc. Camb. Philos. Soc., Volume 11 (1986) no. 3, pp. 427-434 | Article | MR 857720 | Zbl 0612.10010
[6] Finiteness of integral values for the ratio of two linear recurrences, Invent. Math., Volume 149 (2002) no. 2, pp. 431-451 | Article | MR 1918678 | Zbl 1026.11021
[7] Einführung in die Theorie der algebraischen Zahlen und Funktionen, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften, Mathematische Reihe, Birkhäuser, 1963 no. 27 | Zbl 0152.19501
[8] On the Diophantine equation for third order linear recurring sequences, Port. Math. (N.S.), Volume 61 (2004) no. 1, pp. 1-24 | MR 2075259 | Zbl 1114.11030
[9] Decomposable polynomials in second order linear recurrence sequences, Manuscr. Math., Volume 159 (2019) no. 3-4, pp. 321-346 | Article | MR 3959265 | Zbl 07063574
[10] On fewnomials, integral points, and a toric version of Bertini’s theorem, J. Am. Math. Soc., Volume 31 (2018) no. 1, pp. 107-134 | Article | MR 3718452 | Zbl 06802405
[11] Effective bounds for the zeros of linear recurrences in function fields, J. Théor. Nombres Bordeaux, Volume 17 (2005) no. 3, pp. 749-766 | Article | Numdam | MR 2212123 | Zbl 1153.11303
[12] Composite rational functions having a bounded number of zeros and poles, Proc. Am. Math. Soc., Volume 139 (2011) no. 1, pp. 31-38 | Article | MR 2729068 | Zbl 1234.11156
[13] On the Diophantine equation , Monatsh. Math., Volume 137 (2002) no. 3, pp. 173-196 | Zbl 1026.11039
[14] On the Diophantine equation : higher-order recurrences, Trans. Am. Math. Soc., Volume 355 (2003) no. 11, pp. 4657-4681 | Article | Zbl 1026.11040
[15] On the Diophantine equation with , Diophantine Approximation. Festschrift for Wolfgang Schmidt (Developments in Mathematics) Volume 16, Springer, 2008, pp. 199-209 | Article | Zbl 1215.11031
[16] Composite rational functions expressible with few terms, J. Eur. Math. Soc., Volume 14 (2012) no. 1, pp. 175-208 | Article | MR 2862037 | Zbl 1244.12001
[17] When is a polynomial a composition of other polynomials?, Am. Math. Mon., Volume 118 (2011) no. 4, pp. 358-363 | Article | MR 2800347 | Zbl 1233.12002
[18] The classical fields. Structural features of the real and rational numbers, Encyclopedia of Mathematics and Its Applications, Volume 112, Cambridge University Press, 2007 | Zbl 1173.00006
[19] Polynomials with special regard to reducibility, Encyclopedia of Mathematics and Its Applications, Volume 77, Cambridge University Press, 2000 (With an appendix by Umberto Zannier) | MR 1770638 | Zbl 0956.12001
[20] The intersection of recurrence sequences, Acta Arith., Volume 72 (1995) no. 1, pp. 1-44 | Article | MR 1346803 | Zbl 0851.11007
[21] Function Fields and Codes, Universitext, Springer, 1993 | Zbl 0816.14011
[22] Algebraic curves, Dover Publications, 1962 | MR 144897 | Zbl 0103.38202
[23] On the integer solutions of exponential equations in function fields, Ann. Inst. Fourier, Volume 54 (2004) no. 4, pp. 849-874 | Article | Numdam | MR 2111014 | Zbl 1080.11028
[24] On the number of terms of a composite polynomial, Acta Arith., Volume 127 (2007) no. 2, pp. 157-167 | Article | MR 2289981 | Zbl 1161.11003
[25] On composite lacunary polynomials and the proof of a conjecture of Schinzel, Invent. Math., Volume 174 (2008) no. 1, pp. 127-138 | Article | MR 2430978 | Zbl 1177.12004
[26] Addendum to the paper: “On the number of terms of a composite polynomial”, Acta Arith., Volume 140 (2009) no. 1, pp. 93-99 | Article | Zbl 1217.12002