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Harmonic exponential terms are polynomial
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 09 October 2025, pp. 1-6
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- Article
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Let n be a positive integer and f belong to the smallest ring of functions
$\mathbb R^n\to \mathbb R$ that contains all real polynomial functions of n variables and is closed under exponentiation. Then there exists 
$d\in \mathbb N$ such that for all 
$m\in \{0,\dots , n\}$ and 
$c\in \mathbb R^{m}$, if 
$x\mapsto f(c,x)\colon \mathbb R^{n-m}\to \mathbb R$ is harmonic, then it is polynomial of degree at most d. In particular, f is polynomial if it is harmonic.
