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Constant terms in powers of a Laurent polynomial. (English) Zbl 0916.22007
The following is a conjecture of O. Mathieu: Let \(K\) be a connected real compact Lie group. Let \(f\) and \(g\) be \(K\)-finite functions on \(K\). Assume for all \(n \geq 1\) that the constant term of \(f^{n}\) vanishes, i.e. \[ \int_{K} f^{n}(k) \;dk = 0 . \] Then all but finitely many of the constant terms \(\{ \int_{K} f^{n}(k) g(k) \;dk \}\) also vanish. The authors prove this when \(K\) is a real torus with complexification \(T\). The all-important case when \(T = {\mathbf G}_{m}\) involves some clever arguments with Newton polygons.

MSC:
22E30 Analysis on real and complex Lie groups
32A17 Special families of functions of several complex variables
30E99 Miscellaneous topics of analysis in the complex plane
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