algcurves[Siegel] - use Siegel's algorithm for reducing a Riemann matrix
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Calling Sequence
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Siegel(B)
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Description
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A Riemann matrix is a symmetric matrix whose imaginary part is strictly positive definite. In the context of algebraic curves, such a matrix is obtained as a normalized periodmatrix of the algebraic curve.
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A Siegel transformation is a transformation from the canonical basis of the homology of a Riemann surface to a new canonical basis of the homology on the Riemann surface such that:
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1.
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The real part of the new Riemann matrix has entries that are less than or equal to .
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The imaginary part of B is strictly positive definite. Then it can be decomposed as . The columns of T generate a lattice L. Then
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The length of the shortest element of L has a lower bound of ,
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and
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Examples
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![b := Matrix([[.500000055125149+.959847047120791*I, -.500000030191873-.181985134690607*I, .641559302373033+.570916110126901*I], [-.500000001970352-.181985123354161*I, .499999991378705+.866025409210244*I, -.907603726955998-.524005265015065*I], [.641559334390995+.570916144466864*I, -.907603757382622-.524005277968408*I, .549162995386745+1.23866447528125*I]])](/support/helpjp/helpview.aspx?si=7053/file03487/math116.png)
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![Matrix([[.499999991378705+.866025409210244*I, .499999983918887-.181985129022384*I, -.407603726088198-.342020142469353*I], [.499999983918887-.181985129022384*I, -.499999944874851+.959847047120791*I, .141559263256865-.388930919823908*I], [-.407603726088198-.342020142469353*I, .141559263256865-.388930919823908*I, -.233955586252134+1.05667926780828*I]])](/support/helpjp/helpview.aspx?si=7053/file03487/math127.png)
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References
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Deconinck, B., and van Hoeij, M. "Computing Riemann Matrices of Algebraic Curves." Physica D Vol 152-153, (2001): 28-46.
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Siegel, C. L. Topics in Complex Function Theory. Vol. 3. Now York: Wiley, 1973.
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