Abstract
The computation of intersection points of generic tropical hyper-surfaces is a fundamental problem in computational algebraic geometry. An efficient algorithm for solving this problem will be a basic building block in many higher level algorithms for studying tropical varieties, computing mixed volume, enumerating mixed cells, constructing polyhedral homotopies, etc. libtropicon is a library for computing intersection points of generic tropical hyper-surfaces that provides a unified framework where the several conceptually opposite approaches coexist and complement one another. In particular, great efficiency is achieve by the data cross-feeding of the “pivoting” and the “elimination” step — data by-product generated by the pivoting step is selectively saved to bootstrap the elimination step, and vice versa. The core algorithm is designed to be naturally parallel and highly scalable, and the implementation directly supports multi-core architectures, computer clusters, and GPUs based on CUDA or ROCm/OpenCL technology. Many-core architectures such as Intel Xeon Phi are also partially supported. This library also includes interface layers that allows it to be tightly integrated into the existing ecosystem of software in computational algebraic geometry.
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ROCm is AMD’s latest implementation of the OpenCL standard, an open standard for general purpose GPU computation. Currently, only ROCm have been tested. Support for other implementations of OpenCL could be added in the future with minimum changes to the code.
References
Chen, T., Lee, T.L., Li, T.Y.: Mixed cell computation in Hom4PS-3. J. Symbolic Comput. 79, 516–534 (2017)
Gao, T., Li, T.Y., Wu, M.: Algorithm 846. ACM Trans. Math. Softw. 31(4), 555–560 (2005)
Huber, B., Sturmfels, B.: A polyhedral method for solving sparse polynomial systems. Math. Comput. 64(212), 1541–1555 (1995)
Huber, B., Sturmfels, B.: Bernsteins theorem in affine space. Discrete Comput. Geom. 17(2), 137–141 (1997)
Jensen, A.N.: A presentation of the Gfan software. In: Iglesias, A., Takayama, N. (eds.) ICMS 2006. LNCS, vol. 4151, pp. 222–224. Springer, Heidelberg (2006). https://doi.org/10.1007/11832225_21
Lee, T.L., Li, T.Y., Tsai, C.H.: HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method. Computing, 83(2–3), 109–133 (2008)
Li, T., Wang, X.: The BKK root count in \(\mathbb{C}^n\). Math. Comput. Am. Math. Soc. 65(216), 1477–1484 (1996)
Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, vol. 161. American Mathematical Society, Providence (2015)
Malajovich, G.: Computing mixed volume and all mixed cells in quermassintegral time. Found. Comput. Math. 17(5), 1–42 (2016)
Mizutani, T., Takeda, A.: DEMiCs: a software package for computing the mixed volume via dynamic enumeration of all mixed cells. In: Stillman, M., Verschelde, J., Takayama, N. (eds.) Software for Algebraic Geometry, vol. 148. The IMA Volumes in Mathematics and its Applications, pp. 59–79. Springer, New York (2008)
Mizutani, T., Takeda, A., Kojima, M.: Dynamic enumeration of all mixed cells. Discrete Comput. Geom. 37(3), 351–367 (2007)
Rojas, J.M., Wang, X.: Counting affine roots of polynomial systems via pointed Newton polytopes. J. Complex. 12(2), 116–133 (1996)
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Chen, T. (2018). libtropicon: A Scalable Library for Computing Intersection Points of Generic Tropical Hyper-surfaces. In: Davenport, J., Kauers, M., Labahn, G., Urban, J. (eds) Mathematical Software – ICMS 2018. ICMS 2018. Lecture Notes in Computer Science(), vol 10931. Springer, Cham. https://doi.org/10.1007/978-3-319-96418-8_13
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