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algcurves

 Siegel
 use Siegel's algorithm for reducing a Riemann matrix

 Calling Sequence Siegel(B)

Parameters

 B - Riemann matrix

Description

 • 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.
 • 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:
 1 The real part of the new Riemann matrix has entries that are less than or equal to $\left|\frac{1}{2}\right|$.

The imaginary part of B is strictly positive definite. Then it can be decomposed as $\mathrm{\Im }\left(B\right)=\mathrm{transpose}\left(T\right)T$. The columns of T generate a lattice L. Then

 2 The length of the shortest element of L has a lower bound of $\sqrt{\frac{\sqrt{3}}{2}}$,

and

 3 $\mathrm{max}\left(\left|{N}_{i}\right|\right)$ : {${\left|\mathrm{TN}\right|}^{2}\le {R}^{2}$, $N$ an integer vector} has an upper bound depending only on R and g (=dimension of B) (thus not on B).
 • The Siegel(B) command returns a list $\left[\mathrm{s1},\mathrm{s2}\right]$ where $\mathrm{s1}$ is the new Riemann matrix, and $\mathrm{s2}$ is the symplectic transformation matrix on the canonical basis of the homology such that the Riemann matrix in the new basis is $\mathrm{s1}$. If B is a $g$ by $g$ matrix, then $\mathrm{s2}$ is a $2g$ by $2g$ matrix. If $\mathrm{s2}=\mathrm{Matrix}\left(\left[\left[a,b\right],\left[c,d\right]\right]\right)$, where $a,b,c$, and $d$ are $g$ by $g$ matrices, the new Riemann matrix is $\mathrm{s1}=\frac{aB+b}{cB+d}$.

Examples

 > $\mathrm{with}\left(\mathrm{algcurves}\right):$
 > $f≔{y}^{3}-{x}^{9}-2{x}^{3}y:$
 > $b≔\mathrm{periodmatrix}\left(f,x,y,\mathrm{Riemann}\right)$
 ${b}{≔}\left[\begin{array}{ccc}{0.500000055125149}{+}{0.959847047127724}{}{I}& {-0.500000030191873}{-}{0.181985134690607}{}{I}& {0.641559302373033}{+}{0.570916110133834}{}{I}\\ {-0.500000001970352}{-}{0.181985123303614}{}{I}& {0.499999991378705}{+}{0.866025409210244}{}{I}& {-0.907603726955998}{-}{0.524005264964518}{}{I}\\ {0.641559334375792}{+}{0.570916144445057}{}{I}& {-0.907603757382622}{-}{0.524005277968408}{}{I}& {0.549162995371542}{+}{1.23866447525944}{}{I}\end{array}\right]$ (1)
 > $s≔\mathrm{Siegel}\left(b\right):$
 > ${s}_{1}$
 $\left[\begin{array}{ccc}{0.499999991378705}{+}{0.866025409210244}{}{I}& {0.499999983918888}{-}{0.181985128997111}{}{I}& {-0.407603726088198}{-}{0.342020142469353}{}{I}\\ {0.499999983918888}{-}{0.181985128997111}{}{I}& {-0.499999944874851}{+}{0.959847047127724}{}{I}& {0.141559263249263}{-}{0.388930919838278}{}{I}\\ {-0.407603726088198}{-}{0.342020142469353}{}{I}& {0.141559263249263}{-}{0.388930919838278}{}{I}& {-0.233955586252134}{+}{1.05667926780828}{}{I}\end{array}\right]$ (2)
 > ${s}_{2}$
 $\left[\begin{array}{cccccc}{0}& {1}& {0}& {1}& {0}& {1}\\ {1}& {0}& {0}& {-1}& {1}& {-1}\\ {-1}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {1}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}& {1}\end{array}\right]$ (3)

References

 Deconinck, B., and van Hoeij, M. "Computing Riemann Matrices of Algebraic Curves." Physica D Vol 152-153, (2001): 28-46.
 Siegel, C. L. Topics in Complex Function Theory. Vol. 3. Now York: Wiley, 1973.
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