The equation has a global center if and only if a and b satisfy the composition condition a = P(h)h', b = Q(h)h' for polynomials P, Q and trigonometric polynomial h.
Decompositions of Laurent polynomials
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abstract
In the 1920's, Ritt studied the operation of functional composition g o h(x) = g(h(x)) on complex rational functions. In the case of polynomials, he described all the ways in which a polynomial can have multiple `prime factorizations' with respect to this operation. Despite significant effort by Ritt and others, little progress has been made towards solving the analogous problem for rational functions. In this paper we use results of Avanzi--Zannier and Bilu--Tichy to prove analogues of Ritt's results for decompositions of Laurent polynomials, i.e., rational functions with denominator a power of x.
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2025 1verdicts
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Global Centers in Piecewise linear Differential Equations in the Cylinder
The equation has a global center if and only if a and b satisfy the composition condition a = P(h)h', b = Q(h)h' for polynomials P, Q and trigonometric polynomial h.