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An algebraic independence result related to a conjecture of Dixmier on binary form invariants

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Abstract

In order to better understand the structure of classical rings of invariants for binary forms, Dixmier proposed, as a conjectural homogeneous system of parameters, an explicit collection of invariants previously studied by Hilbert. We generalize Dixmier’s collection and show that a particular subfamily is algebraically independent. Our proof relies on showing certain alternating sums of products of binomial coefficients are nonzero. Along the way we provide a very elementary proof à la Racah, namely, only using the Chu–Vandermonde Theorem, for Dixon’s Summation Theorem. We also provide explicit computations of invariants, for the binary octavic, which can serve as ideal introductory examples to Gordan’s 1868 method in classical invariant theory.

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Acknowlegements

For useful discussions or correspondence the author thanks A. Brouwer, J. Chipalkatti, C. Huneke, C. Krattenthaler and J. Van der Jeugt. The author also thanks the anonymous referee for suggesting useful improvements.

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The author declares that he has no conflict of interest.

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  1. Department of Mathematics, University of Virginia, P. O. Box 400137, Charlottesville, VA, 22904-4137, USA

    Abdelmalek Abdesselam

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Abdesselam, A. An algebraic independence result related to a conjecture of Dixmier on binary form invariants. Res Math Sci 6, 26 (2019). https://doi.org/10.1007/s40687-019-0189-x

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