Tropical formulae for summation over a part of [InlineEquation not available: see fulltext.]

Nikita Kalinin; Mikhail Shkolnikov

doi:10.1007/s40879-018-0218-0

Tropical formulae for summation over a part of [InlineEquation not available: see fulltext.]

Nikita Kalinin^*, Mikhail Shkolnikov

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	909-928
Number of pages	20
Journal	European Journal of Mathematics
Volume	5
Issue number	3
DOIs	https://doi.org/10.1007/s40879-018-0218-0
State	Published - 15 Sep 2019
Externally published	Yes

Keywords

Summation
Tropical geometry
[InlineEquation not available: see fulltext.]
π

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10.1007/s40879-018-0218-0

Cite this

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title = "Tropical formulae for summation over a part of [InlineEquation not available: see fulltext.]",

abstract = "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.",

keywords = "Summation, Tropical geometry, [InlineEquation not available: see fulltext.], π",

author = "Nikita Kalinin and Mikhail Shkolnikov",

note = "Publisher Copyright: {\textcopyright} 2018, Springer International Publishing AG, part of Springer Nature.",

year = "2019",

month = sep,

day = "15",

doi = "10.1007/s40879-018-0218-0",

language = "英语",

volume = "5",

pages = "909--928",

journal = "European Journal of Mathematics",

issn = "2199-675X",

publisher = "Springer International Publishing AG",

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

T2 - see fulltext.]

AU - Kalinin, Nikita

AU - Shkolnikov, Mikhail

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

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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 -

Tropical formulae for summation over a part of [InlineEquation not available: see fulltext.]

Abstract

Keywords

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