European Journal of Mathematics

, Volume 5, Issue 3, pp 909–928

# Tropical formulae for summation over a part of Open image in new window

• Nikita Kalinin
• Mikhail Shkolnikov
Research Article

## Abstract

Let , let stand for $$a,b,c,d\in \mathbb Z_{\geqslant 0}$$ such that $$ad-bc=1$$. Define
In other words, we consider the sum of the powers of the triangle inequality defects for the lattice parallelograms (in the first quadrant) of area one.We prove that converges when $$s>1$$ and diverges at $$s=1/2$$. We also prove that
\begin{aligned} \sum \limits _{(a,b,c,d)} \frac{1}{(a+c)^2(b+d)^2(a+b+c+d)^2} = \frac{1}{3}, \end{aligned}
and show a general method to obtain such formulae. The method comes from the consideration of the tropical analogue of the caustic curves, whose moduli give a complete set of continuous invariants on the space of convex domains.

## Keywords

Tropical geometry Summation $$\pi$$

## Mathematics Subject Classification

14T05 14G10 11A55 11A25 11H06

## References

1. 1.
Brugallé, E., Itenberg, I., Mikhalkin, G., Shaw, K.: Brief introduction to tropical geometry. In: Akbulut, S., Auroux, D., Önder, T. (eds.) Proceedings of the Gökova Geometry-Topology Conference 2014, pp. 1–75. International Press, Boston (2015)Google Scholar
2. 2.
Cannas da Silva, A.: Lectures on Symplectic Geometry. Lecture Notes in Mathematics, vol. 1764. Springer, Berlin (2001)Google Scholar
3. 3.
Delzant, T.: Hamiltoniens périodiques et images convexes de l’application moment. Bull. Soc. Math. France 116(3), 315–339 (1988)
4. 4.
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, New York (1979)
5. 5.
Kalinin, N., Shkolnikov, M.: Introduction to tropical series and wave dynamic on them (2017). arXiv:1706.03062 (to appear in Discrete Contin. Dyn. Syst.)
6. 6.
Kalinin, N., Shkolnikov, M.: The number $$\pi$$ and summation by $${S}{L}(2, {\mathbb{Z}})$$. Arnold Math. J.
7. 7.
Shkolnikov, M.: Tropical Curves, Convex Domains, Sandpiles and Aamoebas. Ph.D. thesis, Université de Genève (2017).
8. 8.
Yu, T.Y.: The number of vertices of a tropical curve is bounded by its area. Enseign. Math. 60(3–4), 257–271 (2014)