European Journal of Mathematics

, Volume 5, Issue 3, pp 909–928 | Cite as

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

  • Nikita KalininEmail author
  • Mikhail Shkolnikov
Research Article


Let Open image in new window , let Open image in new window 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 Open image in new window 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.


Tropical geometry Summation Open image in new window \(\pi \) 

Mathematics Subject Classification

14T05 14G10 11A55 11A25 11H06 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsSt. PetersburgRussian Federation
  2. 2.IST AustriaKlosterneuburgAustria

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