The Newton polytope of the discriminant of a quaternary cubic form. (English) Zbl 1454.14144
Summary: We determine the 166104 extremal monomials of the discriminant of a quaternary cubic form. These are in bijection with D-equivalence classes of regular triangulations of the 3-dilated tetrahedron. We describe how to compute these triangulations and their D-equivalence classes in order to arrive at our main result. The computation poses several challenges, such as dealing with the sheer number of triangulations effectively, as well as devising a suitably fast algorithm for computation of a D-equivalence class.
MSC:
| 14Q10 | Computational aspects of algebraic surfaces |
| 52B55 | Computational aspects related to convexity |
Keywords:
triangulations of point configurations; MPTOPCOM; discriminant; quaternary cubic; GKZ vectorsReferences:
| [1] | David Avis - Komei Fukuda,Reverse search for enumeration., Discrete Appl. Math. 65 no. 1-3 (1996), 21-46 (English). · Zbl 0854.68070 |
| [2] | David Avis - Charles Jordan,A parallel framework for reverse search usingmts, 2016, PreprintarXiv:1610.07735. |
| [3] | David Avis - Charles Jordan,mplrs: A scalable parallel vertex/facet enumeration code, Mathematical Programming Computation 10 no. 2 (2018), 267-302. · Zbl 1400.90222 |
| [4] | Dominic Bunnet - Hanieh Keneshlou,Determinantal representations of the cubic discriminant, this volume. · Zbl 1451.14167 |
| [5] | Lasse Collin,XZ Utils, 2019, https://tukaani.org/xz/. |
| [6] | Jes´us A. De Loera - J¨org Rambau - Francisco Santos,Triangulations, Algorithms and Computation in Mathematics, vol. 25, Springer-Verlag, Berlin, 2010, Structures for algorithms and applications. · Zbl 1207.52002 |
| [7] | Ewgenij Gawrilow - Michael Joswig,polymake: a framework for analyzing convex polytopes, Polytopes—combinatorics and computation (Oberwolfach, 1997), DMV Sem., vol. 29, Birkh¨auser, Basel, 2000, pp. 43-73. · Zbl 0960.68182 |
| [8] | I. M. Gel0fand - M. M. Kapranov - A. V. Zelevinsky,Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkh¨auser Boston, Inc., Boston, MA, 1994. · Zbl 0827.14036 |
| [9] | Hiroshi Imai - Tomonari Masada - Fumihiko Takeuchi - Keiko Imai,Enumerating triangulations in general dimensions, Internat. J. Comput. Geom. Appl. 12 no. 6 (2002), 455-480. · Zbl 1045.05006 |
| [10] | Charles Jordan - Michael Joswig - Lars Kastner,Parallel enumeration of triangulations., Electron. J. Comb. 25 no. 3 (2018), research paper p3.6, 27 (English). · Zbl 1393.68175 |
| [11] | Charles Jordan - Michael Joswig - Lars Kastner,mptopcom, version 1.1, Open source software for the parallel enumeration of triangulations (2019), https://polymake.org/mptopcom. |
| [12] | E. J. Nanson,On the eliminant of a set of quadrics, ternary or quaternary., Proc. R. Soc. Edinburgh 22 (1899), 353-358 (English). · JFM 30.0161.09 |
| [13] | J¨org Rambau,TOPCOM: triangulations of point configurations and oriented matroids, Mathematical software (Beijing, 2002), World Sci. Publ., River Edge, NJ, 2002, pp. 330-340. · Zbl 1057.68150 |
| [14] | Bernd Sturmfels - Kristian Ranestad,Twenty-seven questions about the cubic surface, this volume. · Zbl 1253.14055 |
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