Emergent Features in Rational Ehrhart Quasi-Polynomials
Ehrhart Theory presents a tool for the study of integer dilations of integral polytopes (convex polytopes whose vertices all have integer coordinates) via an associated polynomial in the dilation factor. This is then generalized to rational polytopes (allowing vertices to have rational coordinates)...
- Autores:
-
Maldonado Baracaldo, Nicolás
- Tipo de recurso:
- Trabajo de grado de pregrado
- Fecha de publicación:
- 2022
- Institución:
- Universidad de los Andes
- Repositorio:
- Séneca: repositorio Uniandes
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.uniandes.edu.co:1992/64380
- Acceso en línea:
- http://hdl.handle.net/1992/64380
- Palabra clave:
- Ehrhart
Politopo
Polinomio
Cuasi-polinomio
Reticulo
Permutaedro
Matemáticas
- Rights
- openAccess
- License
- Attribution-NoDerivatives 4.0 Internacional
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Emergent Features in Rational Ehrhart Quasi-Polynomials |
| dc.title.alternative.none.fl_str_mv |
Características emergentes en los cuasi-polinomios racionales de Ehrhart |
| title |
Emergent Features in Rational Ehrhart Quasi-Polynomials |
| spellingShingle |
Emergent Features in Rational Ehrhart Quasi-Polynomials Ehrhart Politopo Polinomio Cuasi-polinomio Reticulo Permutaedro Matemáticas |
| title_short |
Emergent Features in Rational Ehrhart Quasi-Polynomials |
| title_full |
Emergent Features in Rational Ehrhart Quasi-Polynomials |
| title_fullStr |
Emergent Features in Rational Ehrhart Quasi-Polynomials |
| title_full_unstemmed |
Emergent Features in Rational Ehrhart Quasi-Polynomials |
| title_sort |
Emergent Features in Rational Ehrhart Quasi-Polynomials |
| dc.creator.fl_str_mv |
Maldonado Baracaldo, Nicolás |
| dc.contributor.advisor.none.fl_str_mv |
Bogart, Tristram |
| dc.contributor.author.none.fl_str_mv |
Maldonado Baracaldo, Nicolás |
| dc.contributor.jury.none.fl_str_mv |
Rau, Johannes |
| dc.subject.keyword.none.fl_str_mv |
Ehrhart Politopo Polinomio Cuasi-polinomio Reticulo Permutaedro |
| topic |
Ehrhart Politopo Polinomio Cuasi-polinomio Reticulo Permutaedro Matemáticas |
| dc.subject.themes.es_CO.fl_str_mv |
Matemáticas |
| description |
Ehrhart Theory presents a tool for the study of integer dilations of integral polytopes (convex polytopes whose vertices all have integer coordinates) via an associated polynomial in the dilation factor. This is then generalized to rational polytopes (allowing vertices to have rational coordinates) yielding a quasi-polynomial in the dilation factor. Linke presented in 2011 a further generalization by allowing the dilation to be rational which again results in a quasi-polynomial in the dilation factor whose coefficients are piecewise polynomial. We herein present our first foray into this field, beginning with a careful review of the literature and all the necessary concepts before tying it all up with a fleshed-out example and the beginnings of an exploration into the form a particular rational Ehrhart quasi-polynomial takes and the information about its associated polytope that may be gleaned from it. |
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2022-11-28 |
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2023-01-31T20:07:52Z |
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2023-01-31T20:07:52Z |
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Federico Ardila, Matthias Beck, and Jodi McWhirter. The arithmetic of Coxeter permutahedra. In: Revista de la Academia Colombiana de Ciencias Exactas, Fisicas y Naturales 44.173 (Dec. 2021), pp. 1152-1166. issn: 23824980. doi: 10.18257/RACCEFYN.1189. Federico Ardila, Anna Schindler, and Andrés R Vindas-Meléndez. The equivariant volumes of the permutahedron. In: Discrete & Computational Geometry 65 (2021), pp. 618-635. doi: https: //doi.org/10.1007/s00454-019-00146-2. url: https://arxiv. org/abs/1803.02377. Federico Ardila, Mariel Supina, and Andrés R Vindas-Meléndez. The equivariant Ehrhart theory of the permutahedron. In: Proceedings of the American Mathematical Society 148 (2020), pp. 5091- 5107. doi: https://doi.org/10.1090/proc/15113. url: https: //arxiv.org/abs/1911.11159. Matthias Beck, Sophia Elia, and Sophie Rehberg. Rational Ehrhart Theory. In: Seminaire Lotharingien de Combinatoire 86B (2022). url: https://arxiv.org/abs/2110.10204. Matthias Beck and Sinai Robins. Computing the Continuous Discretely. 2nd. Undergraduate Texts in Mathematics. New York, NY: Springer New York, 2015. isbn: 978-1-4939-2968-9. doi: 10. 1007/978-1-4939-2969-6. url: http://link.springer.com/ 10.1007/978-1-4939-2969-6. Felix Breuer. An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics. In: Computer Algebra and Polynomials. Springer Cham, May 2014. Chap. 1, pp. 1-29. doi: 10.48550/arxiv.1405.7647. url: https: //arxiv.org/abs/1405.7647v2. Jesús Antonio De Loera. Easy-to-Explain but Hard-to-Solve Problems About Convex Polytopes. 2012. url: http : / / www . math . ucdavis.edu/Ë¿deloera/1. Branko Grünbaum. Convex Polytopes. Ed. by Volker Kaibel, Victor Klee, and Günter M. Ziegler. Graduate Texts in Mathematics. New York, NY: Springer New York, 2003. isbn: 978-0-387- 40409-7. doi: 10.1007/978-1-4613-0019-9. url: http://link. springer.com/10.1007/978-1-4613-0019-9. Eva Linke. Rational Ehrhart quasi-polynomials. In: Journal of Combinatorial Theory, Series A 118.7 (Oct. 2011), pp. 1966-1978. issn: 0097-3165. doi: 10.1016/J.JCTA.2011.03.007. PBS Infinite Series. Proving Pick's Theorem. 2017. url: https:// www.youtube.com/watch?v=bYW1zOMCQno&t=1s. David Sharpe. Rings and Factorization. Cambridge University Press, Aug. 1987. isbn: 9780521330725. doi: 10.1017/CBO9780511565960. url: https://www.cambridge.org/core/product/identifier/ 9780511565960/type/book. N. J. A. Sloane. A138464. 2008. url: https://oeis.org/A138464. Alan Stapledon. Equivariant Ehrhart Theory. In: Advances in Mathematics 226.4 (2011), pp. 3622-3654. doi: https://doi.org/ 10.1016/j.aim.2010.10.019. url: https://arxiv.org/abs/ 1003.5875. Terence Tao. Conversions between standard polynomial bases | What's new. 2019. url: https://terrytao.wordpress.com/2019/04/ 07/conversions-between-standard-polynomial-bases/. Value of Vandermonde Determinant/Formulation 1. url: https : / / proofwiki . org / wiki / Value _ of _ Vandermonde _ Determinant / Formulation_1. Günter M. Ziegler. Lectures on Polytopes. Vol. 152. Graduate Texts in Mathematics. New York, NY: Springer New York, 1995. isbn: 978-0-387-94365-7. doi: 10.1007/978-1-4613-8431-1. url: http://link.springer.com/10.1007/978-1-4613-8431-1. spacematt. Pick's theorem: The wrong, amazing proof. 2021. url: https://www.youtube.com/watch?v=uh-yRNqLpOg. |
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Attribution-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nd/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Bogart, Tristramb29b0027-3c4b-4654-a24e-5db1a4e5456d600Maldonado Baracaldo, Nicolás3c7f02b7-2a60-4525-ad54-58b987855ffd600Rau, Johannes2023-01-31T20:07:52Z2023-01-31T20:07:52Z2022-11-28http://hdl.handle.net/1992/64380instname:Universidad de los Andesreponame:Repositorio Institucional Sénecarepourl:https://repositorio.uniandes.edu.co/Ehrhart Theory presents a tool for the study of integer dilations of integral polytopes (convex polytopes whose vertices all have integer coordinates) via an associated polynomial in the dilation factor. This is then generalized to rational polytopes (allowing vertices to have rational coordinates) yielding a quasi-polynomial in the dilation factor. Linke presented in 2011 a further generalization by allowing the dilation to be rational which again results in a quasi-polynomial in the dilation factor whose coefficients are piecewise polynomial. We herein present our first foray into this field, beginning with a careful review of the literature and all the necessary concepts before tying it all up with a fleshed-out example and the beginnings of an exploration into the form a particular rational Ehrhart quasi-polynomial takes and the information about its associated polytope that may be gleaned from it.La Teoría de Ehrhart presenta una herramienta para el estudio de dilataciones enteras de politopos integrales (politopos convexos cuyos vértices tienen todos coordenadas enteras) a través de un polinomio asociado en el factor de dilatación. Esto luego se generaliza a politopos racionales (permitiendo que los vértices tengan coordenadas racionales), lo cual produce un cuasi-polinomio en el factor de dilatación. Linke presentó en 2011 una generalización adicional al permitir que la dilatación sea racional, lo cual nuevamente da como resultado un cuasi-polinomio en el factor de dilatación cuyos coeficientes son polinomios por partes. Presentamos aquí nuestra primera incursión en este campo, comenzando con una revisión cuidadosa de la literatura y todos los conceptos necesarios antes de concluir con un ejemplo detallado y los comienzos de una exploración en la forma que toma un cuasi-polinomio racional de Ehrhart particular y la información sobre su politopo asociado que se puede extraer de él.MatemáticoPregrado61 páginasapplication/pdfengUniversidad de los AndesMatemáticasFacultad de CienciasDepartamento de MatemáticasEmergent Features in Rational Ehrhart Quasi-PolynomialsCaracterísticas emergentes en los cuasi-polinomios racionales de EhrhartTrabajo de grado - Pregradoinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_7a1fTexthttp://purl.org/redcol/resource_type/TPEhrhartPolitopoPolinomioCuasi-polinomioReticuloPermutaedroMatemáticasFederico Ardila, Matthias Beck, and Jodi McWhirter. The arithmetic of Coxeter permutahedra. In: Revista de la Academia Colombiana de Ciencias Exactas, Fisicas y Naturales 44.173 (Dec. 2021), pp. 1152-1166. issn: 23824980. doi: 10.18257/RACCEFYN.1189.Federico Ardila, Anna Schindler, and Andrés R Vindas-Meléndez. The equivariant volumes of the permutahedron. In: Discrete & Computational Geometry 65 (2021), pp. 618-635. doi: https: //doi.org/10.1007/s00454-019-00146-2. url: https://arxiv. org/abs/1803.02377.Federico Ardila, Mariel Supina, and Andrés R Vindas-Meléndez. The equivariant Ehrhart theory of the permutahedron. In: Proceedings of the American Mathematical Society 148 (2020), pp. 5091- 5107. doi: https://doi.org/10.1090/proc/15113. url: https: //arxiv.org/abs/1911.11159.Matthias Beck, Sophia Elia, and Sophie Rehberg. Rational Ehrhart Theory. In: Seminaire Lotharingien de Combinatoire 86B (2022). url: https://arxiv.org/abs/2110.10204.Matthias Beck and Sinai Robins. Computing the Continuous Discretely. 2nd. Undergraduate Texts in Mathematics. New York, NY: Springer New York, 2015. isbn: 978-1-4939-2968-9. doi: 10. 1007/978-1-4939-2969-6. url: http://link.springer.com/ 10.1007/978-1-4939-2969-6.Felix Breuer. An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics. In: Computer Algebra and Polynomials. Springer Cham, May 2014. Chap. 1, pp. 1-29. doi: 10.48550/arxiv.1405.7647. url: https: //arxiv.org/abs/1405.7647v2.Jesús Antonio De Loera. Easy-to-Explain but Hard-to-Solve Problems About Convex Polytopes. 2012. url: http : / / www . math . ucdavis.edu/Ë¿deloera/1.Branko Grünbaum. Convex Polytopes. Ed. by Volker Kaibel, Victor Klee, and Günter M. Ziegler. Graduate Texts in Mathematics. New York, NY: Springer New York, 2003. isbn: 978-0-387- 40409-7. doi: 10.1007/978-1-4613-0019-9. url: http://link. springer.com/10.1007/978-1-4613-0019-9.Eva Linke. Rational Ehrhart quasi-polynomials. In: Journal of Combinatorial Theory, Series A 118.7 (Oct. 2011), pp. 1966-1978. issn: 0097-3165. doi: 10.1016/J.JCTA.2011.03.007.PBS Infinite Series. Proving Pick's Theorem. 2017. url: https:// www.youtube.com/watch?v=bYW1zOMCQno&t=1s.David Sharpe. Rings and Factorization. Cambridge University Press, Aug. 1987. isbn: 9780521330725. doi: 10.1017/CBO9780511565960. url: https://www.cambridge.org/core/product/identifier/ 9780511565960/type/book.N. J. A. Sloane. A138464. 2008. url: https://oeis.org/A138464.Alan Stapledon. Equivariant Ehrhart Theory. In: Advances in Mathematics 226.4 (2011), pp. 3622-3654. doi: https://doi.org/ 10.1016/j.aim.2010.10.019. url: https://arxiv.org/abs/ 1003.5875.Terence Tao. Conversions between standard polynomial bases | What's new. 2019. url: https://terrytao.wordpress.com/2019/04/ 07/conversions-between-standard-polynomial-bases/.Value of Vandermonde Determinant/Formulation 1. url: https : / / proofwiki . org / wiki / Value _ of _ Vandermonde _ Determinant / Formulation_1.Günter M. Ziegler. Lectures on Polytopes. Vol. 152. Graduate Texts in Mathematics. New York, NY: Springer New York, 1995. isbn: 978-0-387-94365-7. doi: 10.1007/978-1-4613-8431-1. url: http://link.springer.com/10.1007/978-1-4613-8431-1.spacematt. 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