TY - JOUR

T1 - Tropical formulae for summation over a part of [InlineEquation not available

T2 - see fulltext.]

AU - Kalinin, Nikita

AU - Shkolnikov, Mikhail

N1 - Publisher Copyright:
© 2018, Springer International Publishing AG, part of Springer Nature.

PY - 2019/9/15

Y1 - 2019/9/15

N2 - Let [InlineEquation not available: see fulltext.], let [InlineEquation not available: see fulltext.] stand for a, b, c, d∈ Z⩾ 0 such that ad- bc= 1. Define [Equation not available: see fulltext.]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 [InlineEquation not available: see fulltext.] converges when s> 1 and diverges at s= 1 / 2. We also prove that ∑(a,b,c,d)1(a+c)2(b+d)2(a+b+c+d)2=13,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.

AB - Let [InlineEquation not available: see fulltext.], let [InlineEquation not available: see fulltext.] stand for a, b, c, d∈ Z⩾ 0 such that ad- bc= 1. Define [Equation not available: see fulltext.]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 [InlineEquation not available: see fulltext.] converges when s> 1 and diverges at s= 1 / 2. We also prove that ∑(a,b,c,d)1(a+c)2(b+d)2(a+b+c+d)2=13,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.

KW - Summation

KW - Tropical geometry

KW - [InlineEquation not available: see fulltext.]

KW - π

UR - http://www.scopus.com/inward/record.url?scp=85070704285&partnerID=8YFLogxK

U2 - 10.1007/s40879-018-0218-0

DO - 10.1007/s40879-018-0218-0

M3 - 文章

AN - SCOPUS:85070704285

SN - 2199-675X

VL - 5

SP - 909

EP - 928

JO - European Journal of Mathematics

JF - European Journal of Mathematics

IS - 3

ER -