Abstract
We give an explicit solution to the problem of differentiation of hyperelliptic functions in genus 3 case. It is a genus 3 analogue of the result of Frobenius and Stickelberger (J Reine Angew Math 92:311–337, 1882). Our method is based on the series of works by Buchstaber, Enolskii and Leikin. First we introduce a polynomial map \(p:\mathbb {C}^{3g} \rightarrow \mathbb {C}^{2g}\) . Next for \(g = 1,2,3\) we provide 3g polynomial vector fields in \(\mathbb {C}^{3g}\) projectable for p and describe their polynomial Lie algebras. Finally we obtain the corresponding derivations of the field of hyperelliptic functions.
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Acknowledgements
The author thanks Victor M. Buchstaber for fruitful discussions of the results.
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Supported in part by the Young Russian Mathematics award, Royal Society International Exchange grant and the RFBR Project 17-01-00366 A.
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Bunkova, E.Y. Differentiation of genus 3 hyperelliptic functions. European Journal of Mathematics 4, 93–112 (2018). https://doi.org/10.1007/s40879-017-0173-1
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DOI: https://doi.org/10.1007/s40879-017-0173-1
Keywords
- Abelian functions
- Elliptic functions
- Jacobians
- Hyperelliptic curves
- Hyperelliptic functions
- Lie algebra of derivations
- Polynomial vector fields