Study and Characterization of Conflict Sets in the Plane

Authors

DOI:

https://doi.org/10.14393/BEJOM-v7-2026-78534

Keywords:

Geometric locus, Conflict set, Equidistant set, Curve parametrization, Real set germs

Abstract

A geometric locus consists of the description of the set of all points that satisfy a given condition or property. In this context, the conflict set is defined as the geometric locus formed by the points equidistant from two subsets of a metric space. In this work, we investigate the conflict set generated by the interaction between the graph of the function f(x) = kxⁿ, with k ∈ ℝ* and n ∈ ℕ  even, and the x-axis Ox. The analytical characterization of this set is obtained from the combination of a parametrized curve with a half-line. These representations allow for a detailed understanding of the geometric behavior of the sets involved, providing a solid foundation for the study of their local properties and strengthening the field of computational applications. In addition to the geometric analysis, this work contributes to the theoretical development of the notion of germs of real sets, with potential applications in topology and singularity theory.

 

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Author Biographies

  • Edvalter da Silva Sena Filho, State University of Vale do Acaraú

    Edvalter da Silva Sena Filho holds Bachelor's and Master's degrees in Mathematics from the Federal University of Piauí (UFPI) and a Ph.D. in Mathematics, with specialization in Singularity Theory, from the Federal University of Ceará (UFC). He is currently an Associate Professor in the Mathematics Teacher Education Program at the State University Vale do Acaraú (UVA), in Sobral, Ceará, Brazil. His academic work includes teaching, research, and outreach in Mathematics. He is also the author of supplementary mathematics books for basic education, contributing to science communication and to the strengthening of Mathematics education.

  • Leonardo Brandao de Araujo, Federal University of Piauí

    Leonardo Brandão de Araújo was born in Araquém, a district of the municipality of Coreaú, Ceará, located in the northwestern region of the state. He received the Bronze Medal (2017) and the Silver Medal (2018) in Level 3 of the Brazilian Mathematics Olympiad for Public and Private Schools (OBMEP). He holds a Bachelor's degree in Mathematics Education from the State University Vale do Acaraú (UVA). He has participated in undergraduate research programs, including the OBMEP Junior Scientific Initiation Program and the Scientific Initiation and Master's Program (PICME). From 2022 to 2024, he worked as a mathematics teacher in the public school system of the state of Ceará. He is currently pursuing a Master's degree in Mathematics at the Federal University of Piauí (UFPI), supported by a PICME/CAPES scholarship.

  • Davi Ribeiro dos Santos, State University of Vale do Acaraú

    Davi Ribeiro dos Santos is a professor in the Mathematics Licentiate Degree Program at the State University Vale do Acaraú (UVA) in Sobral, Ceará. He holds a Master's degree in Mathematics from the Federal University of Ceará (UFC) and an undergraduate degree in Mathematics from the Catholic University of Pernambuco (UNICAP). He engages in teaching, research, and outreach activities in the fields of Mathematics and Mathematics Education, focusing on the initial and continuing education of teachers. He participates in projects, academic supervision, and institutional initiatives that foster the integration of university and society, contributing to teacher training as well as scientific and educational development.

  • Antonio Aguiar Freitas, State University of Vale do Acaraú

    Antônio Aguiar Freitas was born in Corrente, in the state of Piauí, Brazil. In 2011, he moved to Teresina to pursue a Bachelor's degree in Mathematics Education at the Federal University of Piauí (UFPI), where he began his research career through an undergraduate research program in Differential Geometry. He holds a Master's degree in Differential Geometry from UFPI and a Ph.D. in Mathematics from the Federal University of Ceará (UFC). He previously served as a faculty member at the Federal Institute of Education, Science and Technology of Piauí (IFPI) and is currently a professor in the Mathematics Teacher Education Program at the State University Vale do Acaraú (UVA). His teaching, research, and outreach activities focus on Geometry, teacher education, and technologies applied to Mathematics teaching.

References

[1] Antonio Caminha Muniz Neto. Geometria. 2ª ed. Rio de Janeiro: Sociedade Brasileira de Matemática (SBM), 2022. ISBN: 9788583371854.

[2] Plácido de Jesus da Silvo Leitão Junior. Geometria das Hipersuperfícies de Conflito. Master of Thesis. São Paulo, Brasil, 2002.

[3] J. B. WILKER. Equidistant Sets and their Connectivity Properties. Em: Proceedings of The American Mathematical Society 47 (1975), pp. 446–452.

[4] Dirk Siersma. Properties of conflict sets in the plane. Em: Banach Center Publications 1 (1999), pp. 267–276.

[5] Lev Birbrair e Dirk Siersma. Metric properties of conflict sets. Em: Houston Journal of Mathematics 35 (2009), pp. 73–80.

[6] Mario Ponce e Patricio Santibáñez. On equidistant sets and generalized conics: the old and the new. Em: The American Mathematical Monthly 121 (2014), pp. 18–32.

[7] Leonardo Brandão de Araújo et al. Espaçamento Uniforme: Parametrizando conjuntos de conflitos no plano. Em: XXV Encontro de Iniciação Científica: Desafios, interfaces e contribuições da Ciência para o desenvolvimento sustentável no semiárido cearense. Sobral-CE: UVA, 2024, pp. 511–516. URL: https://drive.google.com/file/d/1mXIZvpKkaFRv2paqrDHCrnmvYdy5_1FR/view.

[8] Jiri Lebl. Singularities and complexity in CR geometry. Thesis. San Diego, USA, 2007.

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Published

2026-07-21

Issue

Vol. 7 (2026): Brazilian Electronic Journal of Mathematics

Section

Pure Mathematics

License

Copyright (c) 2026 Edvalter da Silva Sena Filho, Leonardo Brandao de Araujo, Davi Ribeiro dos Santos, Antonio Aguiar Freitas

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How to Cite

FILHO, Edvalter da Silva Sena; ARAUJO, Leonardo Brandao de; SANTOS, Davi Ribeiro dos; FREITAS, Antonio Aguiar. Study and Characterization of Conflict Sets in the Plane. BRAZILIAN ELECTRONIC JOURNAL OF MATHEMATICS, Uberlândia, Minas Gerais, v. 7, p. 1–23, 2026. DOI: 10.14393/BEJOM-v7-2026-78534. Disponível em: https://seer.ufu.br/index.php/BEJOM/article/view/78534. Acesso em: 22 jul. 2026.