### 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 |
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Original language | English |

Pages (from-to) | 143 - 158 |

Number of pages | 16 |

Journal | Discrete and Computational Geometry |

Volume | 35 (1) |

DOIs | |

Publication status | Published - Jan 2006 |

### Bibliographical note

Publisher: SpringerOther identifier: IDS Number: 988KX

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## Cite this

Iosevich, A., Konyagin, S., Rudnev, M., & Ten, VV. (2006). Combinatorial complexity of convex sequences.

*Discrete and Computational Geometry*,*35 (1)*, 143 - 158. https://doi.org/10.1007/s00454-005-1194-y