Abstract
Addition of points on the tropical Hesse curve is realized via its intersections with two tropical lines. Then the addition formula for the points on the curve is reduced from the one for the level-three theta functions through the ultradiscretization procedure. In addition, a tropical analogue of the Hessian group \(G_{216}\), the group of linear automorphisms acting on the Hesse pencil, is investigated; it is shown that the dihedral group \(\mathcal {D}_3\) of degree three is the group of linear automorphisms acting on the tropical Hesse pencil.
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Notes
The Hessian curve of a nonsingular cubic curve E is the plane cubic curve defined by the equation \({\mathrm{He}}(F) = 0\), where \({\mathrm{He}}(F)\) is the determinant of the matrix of the second partial derivatives of the defining polynomial F of E.
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This work was partially supported by JSPS KAKENHI Grant Numbers 22740100 and 26400107. The author is grateful to an anonymous referee for his/her helpful comments and suggestions.
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Nobe, A. Group actions on the tropical Hesse pencil. Japan J. Indust. Appl. Math. 33, 537–556 (2016). https://doi.org/10.1007/s13160-016-0226-8
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DOI: https://doi.org/10.1007/s13160-016-0226-8