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A \(t_k\) Inequality for Arrangements of Pseudolines

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Abstract

Consider arrangements of n pseudolines in the real projective plane. Let \(t_k\) denote the number of intersection points where exactly k pseudolines are incident. We present a new combinatorial inequality:

$$\begin{aligned} t_2+1.5t_3\ge 8+\sum _{k\ge 4}(2k-7.5)t_k, \end{aligned}$$

which holds if no more than \(n-3\) pseudolines intersect at one point. It looks similar but is unrelated to the Hirzebruch inequality for arrangements of complex lines in the complex projective plane. Based on this linear inequality, we construct lower bounds for the number of regions via n and the maximal number of (pseudo)lines passing through one point.

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Acknowledgments

I am grateful to the late V. I. Arnold and to A. B. Skopenkov for their interest in this work. I am deeply grateful to the referees and E. G. Puniskij for improvements to the structure and language of the manuscript.

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  1. WorldQuant Research (Eurasia), 1035, Smolenskaya sq. 3, Moscow, Russia

    I. N. Shnurnikov

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  1. I. N. Shnurnikov
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Correspondence to I. N. Shnurnikov.

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Shnurnikov, I.N. A \(t_k\) Inequality for Arrangements of Pseudolines. Discrete Comput Geom 55, 284–295 (2016). https://doi.org/10.1007/s00454-015-9744-4

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  • DOI: https://doi.org/10.1007/s00454-015-9744-4

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