Abstract:
The Newton polytope of the resultant, or resultant polytope, characterizes the resultant polynomial more precisely than total degree. The combinatorics of resultant polytopes are known in the Sylvester case [5] and up to dimension 3 [9]. We extend this work by studying the combinatorial characterization of 4-dimensional resultant polytopes, which show a greater diversity and involve computational and combinatorial challenges. In particular, our experiments, based on software respol for computing resultant polytopes, establish lower bounds on the maximal number of faces. By studying mixed subdivisions, we obtain tight upper bounds on the maximal number of facets and ridges, thus arriving at the following maximal f-vector: (22, 66, 66, 22), i.e. vector of face cardinalities. Certain general features emerge, such as the symmetry of the maximal f-vector, which are intriguing but still under investigation. We establish a result of independent interest, namely that the f-vector is maximized when the input supports are sufficiently generic, namely full dimensional and without parallel edges. Lastly, we offer a classification result of all possible 4-dimensional resultant polytopes. Copyright 2013 ACM.
Registro:
| Documento: |
Conferencia
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| Título: | Combinatorics of 4-dimensional resultant polytopes |
| Autor: | Dickenstein, A.; Emiris, I.Z.; Fisikopoulos, V. |
| Ciudad: | Boston, MA |
| Filiación: | Dto. de Matemática, IMAS-CONICET, Universidad de Buenos Aires, Argentina
Department of Informatics and Telecommunications, University of Athens, Greece
|
| Palabras clave: | F-vector; Mixed subdivision; Resultant; Secondary polytope; Cardinalities; Classification results; Combinatorics; F vectors; Mixed subdivision; Newton polytopes; Polytopes; Resultant; Algebra; Combinatorial mathematics; Vectors; Topology |
| Año: | 2013
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| Página de inicio: | 173
|
| Página de fin: | 180
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| DOI: |
http://dx.doi.org/10.1145/2465506.2465937 |
| Título revista: | 38th International Symposium on Symbolic and Algebraic Computation, ISSAC 2013
|
| Título revista abreviado: | Proc Int Symp Symbol Algebraic Comput ISSAC
|
| Registro: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_97814503_v_n_p173_Dickenstein |
Referencias:
- Cattani, E., Cueto, M.A., Dickenstein, A., Rocco, S.D., Sturmfels, B., Mixed discriminants (2011) Math. Z., 2013., , to appear; also in ArXiv
- Loera, J.D., Rambau, J.A., Santos, F., (2010) Triangulations: Structures for Algorithms and Applications, 25. , Springer-Verlag
- Emiris, I.Z., Fisikopoulos, V., Konaxis, C., Pẽnaranda, L., An output-sensitive algorithm for computing projections of resultant polytopes (2012) Proc. ACM Symp. Comp. Geometry, pp. 179-188
- Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V., (1994) Discriminants, Resultants and Multidimensional Determinants, , Birkḧauser, Boston
- Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V., Newton polytopes of the classical resultant and discriminant (1990) Advances in Math., 84, pp. 237-254
- Jensen, A., Yu, J., Computing tropical resultants (2011) J. Algebra., 2013, , to appear; also in ArXiv
- Kalai, G., Rigidity and the lower bound theorem 1 (1987) Inventiones Mathematicae, 88, pp. 125-151
- Rambau, J., TOPCOM: Triangulations of point configurations and oriented matroids (2002) Proc. Intern. Congress Math. Software, pp. 330-340
- Sturmfels, B., On the newton polytope of the resultant (1994) J. Algebr. Combin, 3, pp. 207-236
- Ziegler, G.M., (1995) Lectures on Polytopes, , SpringerA4 - Association for Computing Machinery and Its Special; Interest Group on Symbolic Manipulation (ACM SIGSAM)
Citas:
---------- APA ----------
Dickenstein, A., Emiris, I.Z. & Fisikopoulos, V.
(2013)
. Combinatorics of 4-dimensional resultant polytopes. 38th International Symposium on Symbolic and Algebraic Computation, ISSAC 2013, 173-180.
http://dx.doi.org/10.1145/2465506.2465937---------- CHICAGO ----------
Dickenstein, A., Emiris, I.Z., Fisikopoulos, V.
"Combinatorics of 4-dimensional resultant polytopes"
. 38th International Symposium on Symbolic and Algebraic Computation, ISSAC 2013
(2013) : 173-180.
http://dx.doi.org/10.1145/2465506.2465937---------- MLA ----------
Dickenstein, A., Emiris, I.Z., Fisikopoulos, V.
"Combinatorics of 4-dimensional resultant polytopes"
. 38th International Symposium on Symbolic and Algebraic Computation, ISSAC 2013, 2013, pp. 173-180.
http://dx.doi.org/10.1145/2465506.2465937---------- VANCOUVER ----------
Dickenstein, A., Emiris, I.Z., Fisikopoulos, V. Combinatorics of 4-dimensional resultant polytopes. Proc Int Symp Symbol Algebraic Comput ISSAC. 2013:173-180.
http://dx.doi.org/10.1145/2465506.2465937