The characteristic polynomials of semigeneric threshold arrangements

Ruimei Gao; Yuyu Wang; Ruimei Gao; Yuyu Wang

doi:10.3934/math.20231462

AIMS Mathematics

2023, Volume 8, Issue 12: 28569-28581. doi: 10.3934/math.20231462

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Research article

The characteristic polynomials of semigeneric threshold arrangements

Ruimei Gao ^,,
Yuyu Wang

School of Mathematics and Statistics, Changchun University of Science and Technology, Changchun 130022, China

Received: 29 July 2023 Revised: 08 October 2023 Accepted: 12 October 2023 Published: 20 October 2023
MSC : 52C35, 32S22

The semigeneric threshold arrangement is the hyperplane arrangement defined by $ x_i+x_j = a_{i} $, where $ 1 \leq i < j \leq n $ and $ a_i\text{'s} $ are generic elements. In this paper, we obtain a necessary and sufficient condition for subarrangements of the semigeneric threshold arrangement to be central from the perspective of simple graphs. Combining it with Whitney's theorem, we provide a formula for the characteristic polynomials of the semigeneric threshold arrangement and its subarrangements.
- semigeneric threshold arrangement,
- characteristic polynomial,
- extended even cycle,
- Whitney's theorem
Citation: Ruimei Gao, Yuyu Wang. The characteristic polynomials of semigeneric threshold arrangements[J]. AIMS Mathematics, 2023, 8(12): 28569-28581. doi: 10.3934/math.20231462

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Abstract

The semigeneric threshold arrangement is the hyperplane arrangement defined by $ x_i+x_j = a_{i} $, where $ 1 \leq i < j \leq n $ and $ a_i\text{'s} $ are generic elements. In this paper, we obtain a necessary and sufficient condition for subarrangements of the semigeneric threshold arrangement to be central from the perspective of simple graphs. Combining it with Whitney's theorem, we provide a formula for the characteristic polynomials of the semigeneric threshold arrangement and its subarrangements.

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