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Covariant algebra of the binary nonic and the binary decimic. (English) Zbl 1366.13006
Bassa, Alp (ed.) et al., Arithmetic, geometry, cryptography and coding theory. 15th international conference on arithmetic, geometry, cryptography, and coding theory (AGCT), CIRM, Luminy, France, May 18–22, 2015. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2810-5/pbk; 978-1-4704-3745-9/ebook). Contemporary Mathematics 686, 65-91 (2017).
The present article is devoted to the covariants of binary forms, which is a part of the classical invariant theory. Let \(V\) be a space of binary forms and denote by \(\mathbf{Inv}(V)\) the invariant algebra \(\mathbb C[V]^{\text{SL}_2(\mathbb C)}\). According to the famous result by Gordan (extended by Hilbert for any linear reductive group), the invariant algebra \(\mathbb C[V]^{\text{SL}_2(\mathbb C)}\) is finitely generated. Similarly, the covariant algebra \(\mathbf{Cov}(V)=\mathbb C[V\oplus \mathbb C^2]^{\text{SL}_2(\mathbb C)}\) is finitely generated. The authors of this article show that there exists a minimal covariant basis with 476 generators for the binary nonic (Theorem 24), respectively with 510 generators for the binary decimic (Theorem 25). These results were only knows as conjectures so far. The computations rely on some important improvements on Gordan’s algorithm.
For the entire collection see [Zbl 1364.11004].

MSC:
13A50 Actions of groups on commutative rings; invariant theory
13C99 Theory of modules and ideals in commutative rings
14Q99 Computational aspects in algebraic geometry
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