Combinatorial complexity of convex sequences

A Iosevich; S Konyagin; M Rudnev; VV Ten

doi:10.1007/s00454-005-1194-y

Combinatorial complexity of convex sequences

A Iosevich, S Konyagin, M Rudnev, VV Ten

Research output: Contribution to journal › Article (Academic Journal) › peer-review

18 Citations (Scopus)

Abstract

We show that the equation s(i1) +s(i2) +... +s(id) =si(d+1) + - +si(2d) has O (N2d-2+2-d+1) solutions for any strictly convex sequence {s(i)}(i=1)(N) without any additional arithmetic assumptions. The proof is based on weighted incidence theory and an inductive procedure which allows us to deal with higher-dimensional interactions effectively. The terminology "combinatorial complexity" is borrowed from [CES+] where much of our higher-dimensional incidence theoretic motivation comes from.

Translated title of the contribution	Combinatorial complexity of convex sequences
Original language	English
Pages (from-to)	143 - 158
Number of pages	16
Journal	Discrete and Computational Geometry
Volume	35 (1)
DOIs	https://doi.org/10.1007/s00454-005-1194-y
Publication status	Published - Jan 2006

Bibliographical note

Publisher: Springer
Other identifier: IDS Number: 988KX

Access to Document

10.1007/s00454-005-1194-y

Handle.net

Persistent link

Cite this

@article{5d0993dc46204cf199fd31b34986e192,

title = "Combinatorial complexity of convex sequences",

abstract = "We show that the equation s(i1) +s(i2) +... +s(id) =si(d+1) + - +si(2d) has O (N2d-2+2-d+1) solutions for any strictly convex sequence {s(i)}(i=1)(N) without any additional arithmetic assumptions. The proof is based on weighted incidence theory and an inductive procedure which allows us to deal with higher-dimensional interactions effectively. The terminology {"}combinatorial complexity{"} is borrowed from [CES+] where much of our higher-dimensional incidence theoretic motivation comes from.",

author = "A Iosevich and S Konyagin and M Rudnev and VV Ten",

note = "Publisher: Springer Other identifier: IDS Number: 988KX ",

year = "2006",

month = jan,

doi = "10.1007/s00454-005-1194-y",

language = "English",

volume = "35 (1)",

pages = "143 -- 158",

journal = "Discrete and Computational Geometry",

issn = "0179-5376",

publisher = "Springer, New York, NY",

}

TY - JOUR

T1 - Combinatorial complexity of convex sequences

AU - Iosevich, A

AU - Konyagin, S

AU - Rudnev, M

AU - Ten, VV

N1 - Publisher: Springer Other identifier: IDS Number: 988KX

PY - 2006/1

Y1 - 2006/1

N2 - We show that the equation s(i1) +s(i2) +... +s(id) =si(d+1) + - +si(2d) has O (N2d-2+2-d+1) solutions for any strictly convex sequence {s(i)}(i=1)(N) without any additional arithmetic assumptions. The proof is based on weighted incidence theory and an inductive procedure which allows us to deal with higher-dimensional interactions effectively. The terminology "combinatorial complexity" is borrowed from [CES+] where much of our higher-dimensional incidence theoretic motivation comes from.

AB - We show that the equation s(i1) +s(i2) +... +s(id) =si(d+1) + - +si(2d) has O (N2d-2+2-d+1) solutions for any strictly convex sequence {s(i)}(i=1)(N) without any additional arithmetic assumptions. The proof is based on weighted incidence theory and an inductive procedure which allows us to deal with higher-dimensional interactions effectively. The terminology "combinatorial complexity" is borrowed from [CES+] where much of our higher-dimensional incidence theoretic motivation comes from.

U2 - 10.1007/s00454-005-1194-y

DO - 10.1007/s00454-005-1194-y

M3 - Article (Academic Journal)

SN - 0179-5376

VL - 35 (1)

SP - 143

EP - 158

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

ER -

Combinatorial complexity of convex sequences

Abstract

Bibliographical note

Access to Document

Handle.net

Fingerprint

Cite this