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The symmetric function theorem via the multivariate Faà di Bruno formula

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Abstract

The symmetric function theorem states that a polynomial that is invariant under permutation of variables, is a polynomial in the elementary symmetric polynomials. We deduce this classical result, in the analytic setting, from the multivariate Faà di Bruno formula. In two variables, this allows us to completely determine all coefficients that occur in the inductive equations.

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Acknowledgements

The author would like to thank Léo Jiminez for discussing the interesting combinatorial phenomenon we encountered, especially for writing down a formula for the map \(\phi \).

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  1. McMaster University, 1280 Main Street West, Hamilton, ON, L8S4L8, Canada

    Siegfried Van Hille

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  1. Siegfried Van Hille
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Van Hille is the only author of this article and has written all of it.

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Correspondence to Siegfried Van Hille.

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Van Hille, S. The symmetric function theorem via the multivariate Faà di Bruno formula. European Journal of Mathematics 9, 81 (2023). https://doi.org/10.1007/s40879-023-00679-0

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  • DOI: https://doi.org/10.1007/s40879-023-00679-0

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