On decompositions of trigonometric polynomials. (English) Zbl 1372.30028
Let \(p=p(\cos x, \sin x)\), \(q=q(\cos x, \sin x)\) be trigonometric polynomials over \(\mathbb{R}\), that is, the elements of the ring generated by the functions \(\cos x\), \(\sin x\). The main problem under consideration in the present paper is the description of solutions of the equation
\[
P_1 \circ w_1=P_2 \circ w_2,
\]
where \(w_1, w_2\) are trigonometric polynomials, and \(P_1, P_2\) are real algebraic polynomials. The symbol \(\circ\) denotes the composition of functions.
As it turns out, such general equation can be reduced to a few particular ones, for instance, \[ \tilde P_1 \circ W_1=T_m \circ T_n, \] where \(\tilde P_1, W_1\) are real algebraic polynomials, \(T_n\) are the standard Chebyshev polynomials of the first kind and \(\mathrm{GCD}(m,n)=1\).
As it turns out, such general equation can be reduced to a few particular ones, for instance, \[ \tilde P_1 \circ W_1=T_m \circ T_n, \] where \(\tilde P_1, W_1\) are real algebraic polynomials, \(T_n\) are the standard Chebyshev polynomials of the first kind and \(\mathrm{GCD}(m,n)=1\).
Reviewer: Leonid Golinskii (Kharkov)
MSC:
| 30E05 | Moment problems and interpolation problems in the complex plane |
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
| 30C10 | Polynomials and rational functions of one complex variable |
Keywords:
decomposition equations; trigonometric polynomials; moment problem; Abel equation; Chebyshev polynomialsReferences:
| [1] | A. Álvarez, J. L. Bravo and C. Christopher, On the trigonometric moment problem, Ergodic Theory Dynam. Systems 34 (2014), 1-20. · Zbl 1291.37085 · doi:10.1017/etds.2012.143 |
| [2] | M. Briskin, F. Pakovich and Y. Yomdin, Algebraic geometry of the center-focus problem for Abel differential equations, Ergodic Theory Dynam. Systems 36 (2016), 714-744. · Zbl 1362.37055 · doi:10.1017/etds.2014.94 |
| [3] | M. Briskin, N. Roytvarf and Y. Yomdin, Center conditions at infinity for Abel differential equations, Ann. of Math. (2) 172 (2010), 437-483. · Zbl 1216.34025 · doi:10.4007/annals.2010.172.437 |
| [4] | L. Cherkas, Number of limit cycles of an autonomous second-order system, Differentsial’nye uravneniya 12 (1976), 944-944. · Zbl 0022.32301 |
| [5] | A. Cima, A. Gasull and F. Ma˜nosas, A simple solution of some composition conjectures for Abel equations, J. Math. Anal. Appl. 398 (2013), 477-486. · Zbl 1269.34035 · doi:10.1016/j.jmaa.2012.09.006 |
| [6] | H. T. Engstrom, Polynomial substitutions, Amer. J. Math. 63 (1941), 249-255. · Zbl 0025.10403 · doi:10.2307/2371520 |
| [7] | J. Giné, M. Grau and J. Llibre, Universal centres and composition conditions, Proc. Lond. Math. Soc. (3) 106 (2013), 481-507. · Zbl 1277.34034 · doi:10.1112/plms/pds050 |
| [8] | M. Muzychuk and F. Pakovich, Jordan-Hölder theorem for imprimitivity systems and maximal decompositions of rational functions, Proc. Lond. Math. Soc. (3) 102 (2011), 1-24. · Zbl 1241.20031 · doi:10.1112/plms/pdq009 |
| [9] | F. Pakovich, Prime and composite Laurent polynomials, Bull. Sci. Math. 133 (2009), 693-732. · Zbl 1205.30025 · doi:10.1016/j.bulsci.2009.06.003 |
| [10] | F. Pakovich, On the equation P(f) = Q(g), where P, Q are polynomials and f, g are entire functions, Amer. J. Math. 132 (2010), 1591-1607. · Zbl 1210.39025 |
| [11] | F. Pakovich, Generalized ‘second Ritt theorem’ and explicit solution of the polynomial moment problem, Compos. Math. 149 (2013), 705-728. · Zbl 1273.30024 · doi:10.1112/S0010437X12000620 |
| [12] | F. Pakovich, On rational functions orthogonal to all powers of a given rational function on a curve, Mosc. Math. J. 13 (2013), 693-731. · Zbl 1297.30065 |
| [13] | F. Pakovich, Weak and strong composition conditions for the Abel differential equation, Bull. Sci. Math. 138 (2014), 993-998. · Zbl 1315.34040 · doi:10.1016/j.bulsci.2014.06.001 |
| [14] | F. Pakovich and M. Muzychuk, Solution of the polynomial moment problem, Proc. Lond. Math. Soc. (3) 99 (2009), 633-657. · Zbl 1177.30046 · doi:10.1112/plms/pdp010 |
| [15] | Pakovich, F.; Pech, C.; Zvonkin, A. K., Laurent polynomial moment problem: a case study, in Dynamical numbers—interplay between dynamical systems and number theory, 177-194 (2010) · Zbl 1223.30010 |
| [16] | J. F. Ritt, Prime and composite polynomials, Trans. Amer. Math. Soc. 23 (1922), 51-66. · JFM 48.0079.01 · doi:10.1090/S0002-9947-1922-1501189-9 |
| [17] | A. Schinzel, Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications, Vol. 77, Cambridge University Press, Cambridge, 2000, With an appendix by Umberto Zannier. · Zbl 0956.12001 · doi:10.1017/CBO9780511542916 |
| [18] | U. Zannier, Ritt’s second theorem in arbitrary characteristic, J. Reine Angew. Math. 445 (1993), 175-203. · Zbl 0786.12001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
