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.

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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 ${ℂ}^2$, 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 $y^2=P(x)$, where $P$ 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

… ►This Riemann matrix originates from the Riemann surface represented by the algebraic curve ${\mu }^3-{\lambda }^7+2{\lambda }^3\mu =0$; compare §21.7(i). … ►

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Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: $\mathrm{\Re }\widehat{\theta }\left(x+\mathrm{i}y,0,0\right|{𝛀}_2)$, $0\le x\le 1$, $0\le y\le 3$. This Riemann matrix originates from the genus 3 Riemann surface represented by the algebraic curve ${\mu }^3+2\mu -{\lambda }^4=0$; compare §21.7(i). Magnify 3D Help

5: 23.20 Mathematical Applications

… ►An algebraic curve that can be put either into the form …

6: Bibliography D

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Computing Riemann matrices of algebraic curves. Phys. D 152/153, pp. 28–46.

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External Links:

ISSN 0167-2789, Document, MathReview (John B. Little), ZentralBlatt

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3rd item.

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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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External Links:

ISSN 0747-7171, Document, MathReview (Gabino González Díez), ZentralBlatt

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1st item.

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BibTeX

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8: Bibliography B

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Plane Algebraic Curves. Birkhäuser Verlag, Basel.

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Notes:

Translated from the German by John Stillwell

External Links:

ISBN 3-7643-1769-8, MathReview, ZentralBlatt

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§21.7(i).

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BibTeX

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9: 37.2 General Orthogonal Polynomials of Two Variables

… ►The zero set of a polynomial of two variables may consist of algebraic curves, making things complicated. …

10: 31.8 Solutions via Quadratures

… ►The variables $\lambda$ and $\nu$ are two coordinates of the associated hyperelliptic (spectral) curve $\mathrm{\Gamma }:{\nu }^2={\prod }_{j=1}^{2g+1}(\lambda -{\lambda }_j)$. … ►For $𝐦=(m_0,0,0,0)$, these solutions reduce to Hermite’s solutions (Whittaker and Watson (1927, §23.7)) of the Lamé equation in its algebraic form. The curve $\mathrm{\Gamma }$ reflects the finite-gap property of Equation (31.2.1) when the exponent parameters satisfy (31.8.1) for $m_j\in \mathbb{Z}$. …