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Computing tropical curves via homotopy continuation. (English) Zbl 1350.14045

For an ideal \(I\), the tropical variety is the limit \(\lim_{t\to\infty}\frac{1}{t}\mathcal A(I)\) where \(\mathcal A(I)\) is the image of the variety \(V(I)\) under the map \(\log|\cdot|\). \(\mathcal A(I)\) is called the amoeba of \(V(I)\).
The authors present a construction of the tropicalization of a complex curve using numerical methods, namely homotopy approximation. It was first used for tropical geometry in [D. Adrovic and J. Verschelde, in: Proceedings of the 37th international symposium on symbolic and algebraic computation, ISSAC 2012, Grenoble, France, July 22–25, 2012. New York, NY: Association for Computing Machinery (ACM). 20–27 (2012; Zbl 1323.68578)] and [B. Huber and B. Sturmfels, Math. Comput. 64, No. 212, 1541–1555 (1995; Zbl 0849.65030)] on polyhedral homotopies.
By intersection with hyperplanes it is possible to reduce the problem of construction a tropical variety to the problem of construction a tropical curve. The strategy of the authors is as follows.
We intersect a tropical curve with a hyperplane. We can approximately compute the intersection points. Moving the plane we get an approximation of the direction of the edges of the curve. These arguments, using balancing condition and a priori degree of the curve, can be rephrased in terms of an algorithm.
The paper contains links to the software, examples of computations for \(A\)-polynomials of knots, and time comparison with Gfan (another way to compute a tropical variety is to use Gröbner basis computation, which is implemented in Gfan).

MSC:

14T05 Tropical geometry (MSC2010)
14Q05 Computational aspects of algebraic curves
57M25 Knots and links in the \(3\)-sphere (MSC2010)
65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
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References:

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