Abstract
At present, a plane algebraic curve can be parametrized in the following two cases: if its genus is equal to 0 or 1 and if it has a large group of birational automorphisms. Here we propose a new polyhedron method (involving a polyhedron called a Hadamard polyhedron by the author), which allows us to divide the space ℝ2 or ℂ2 into pieces in each of which the polynomial specifying the curve is sufficiently well approximated by its truncated polynomial, which often defines the parametrized curve. This approximate parametrization in a piece can be refined by means of the Newton method. Thus, an arbitrarily exact piecewise parametrization of the original curve can be obtained.
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References
K. Fukuda, “Exact algorithms and software in optimization and polyhedral computation,” in XXI International Symposium on Symbolic and Algebraic Computations (ACM, New York, 2008), pp. 333-334.
C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The Quickhull algorithm for convex hulls,,” ACM Trans. Math. Software 22 4), 469–483 (1996).
Higher Transcendental Functions, Vol. 3: Elliptic and Modular Functions, Lame and Mathieu Functions (McGraw-Hill, New York-Toronto-London, 1955; Nauka, Moscow, 1967).
Yu. V. Brezhnev, “On uniformization of algebraic curves,,” Moscow Math. J. 8 2), 233–271 (2008).
Yu. V. Brezhnev, “On uniformization of Burnside’s curve y 2 = x5 — x,” J. Math. Phys. 50 (103519) (2009).
Yu. V. Brezhnev, “The sixth Painlevé transcendent and uniformization of algebraic curves,” J. Differential Equations 260 (3), 2507–2556(2016).
E. D. Belokolos and V. Z. Enolskii, “Reduction of abelian functions and algebraically integrable systems. I,” J. Math. Sci. (New York) 106 (6), 3395–3486 (2001).
E. D. Belokolos and V. Z. Enolskii, “Reduction of abelian functions and algebraically integrable systems. II,” J. Math. Sci. (New York) 108 3), 295–374 (2002).
Y Ônishi, “Complex multiplication formulae for hyperelliptic curves of genus three,,” Tokyo J. Math. 21 2), 381–431 (1998).
A. I. Aptekarev, D. N. Tulyakov, and M. L. Yattselev, “On the parametrization of a certain algebraic curve of genus 2,,” Mat. Zametki 98 5), 782–785 (2015) [Math. Notes 98 5), 843-846 (2015)].
A. I. Aptekarev, D. N. Toulyakov, and W. Van Assche, “Hyperelliptic uniformization of algebraic curves of third order,,” J. Comput. Appl. Math. 284, 1–14 2002.
R. J. Walker, Algebraic Curves (Dover Publ., New York, 1962; (“Librokom,” Moscow, 2009).
J. C. Eilbeck, V. Z. Enolski, S. Matsutani, Y. Onishi, and E. Previato, “Abelian functions for trigonal curves of genus three,” Int. Math. Res. Not. IMRN, No. 1 (2008), Art. ID rnm140.
J. C. Eilbeck, Weierstrass Functions for Higher Genus Curves, http://www.ma.hw.ac.uk/Weierstrass (2011).
A. D. Bryuno, “Algorithms for solving an algebraic equation,,” Programming and Computer Software 44 6), 533–545 (2018).uri191220000005014
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Russian Text © The Author(s), 2019, published in Matematicheskie Zametki, 2019, Vol. 106, No. 6, pp. 837-847.
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Bryuno, A.D. On the Parametrization of an Algebraic Curve. Math Notes 106, 885–893 (2019). https://doi.org/10.1134/S0001434619110233
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DOI: https://doi.org/10.1134/S0001434619110233