algebraic curves
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1: 21.10 Methods of Computation
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Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.
Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.
2: 21.7 Riemann Surfaces
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§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces
… ►Belokolos et al. (1994, §2.1)), they are obtainable from plane algebraic curves (Springer (1957), or Riemann (1851)). …Equation (21.7.1) determines a plane algebraic curve in , which is made compact by adding its points at infinity. … ►§21.7(iii) Frobenius’ Identity
… ►These are Riemann surfaces that may be obtained from algebraic curves of the form …3: 22.18 Mathematical Applications
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§22.18(i) Lengths and Parametrization of Plane Curves
… ►Algebraic curves of the form , where is a nonsingular polynomial of degree 3 or 4 (see McKean and Moll (1999, §1.10)), are elliptic curves, which are also considered in §23.20(ii). …The theory of elliptic functions brings together complex analysis, algebraic curves, number theory, and geometry: Lang (1987), Siegel (1988), and Serre (1973). …4: 21.4 Graphics
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►This Riemann matrix originates from the Riemann surface represented by the algebraic curve
; compare §21.7(i).
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5: 23.20 Mathematical Applications
6: Bibliography D
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Computing Riemann matrices of algebraic curves.
Phys. D 152/153, pp. 28–46.
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7: Bibliography G
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Riemann surfaces, plane algebraic curves and their period matrices.
J. Symbolic Comput. 26 (6), pp. 789–803.
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8: Bibliography B
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Plane Algebraic Curves.
Birkhäuser Verlag, Basel.
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9: 31.8 Solutions via Quadratures
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►The variables and are two coordinates of the associated hyperelliptic (spectral) curve
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►For , these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form.
The curve
reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for .
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10: Bibliography T
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Algebraic transformations of hypergeometric functions and automorphic forms on Shimura curves.
Trans. Amer. Math. Soc. 365 (12), pp. 6697–6729.
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