The AbelMap command computes the Abel map between two points P and P_0 on a Riemann surface R of genus g, that is a g-tuple of complex numbers. The jth element of the Abel map is the integral of the jth normalized holomorphic differential integrated along a path from P to P_0.

The Riemann surface is entered as F; an irreducible, square-free polynomial in x and y. Floating point numbers are not allowed as coefficients of F. Algebraic numbers are allowed. Curves of arbitrary finite genus with arbitrary singularities are allowed.

The points P and P_0 are entered as $\&lsqb;x=a+b{t}^r,y=\left(\mathrm{Laurent\; series}\mathrm{in}t\right)\&rsqb;$, where a and b are constants, and r is an integer. If r < 0, that is, if entering one of the points for $x=\mathrm{\infty }$, then a = 0.

The differentials are normalized such that the jth differential integrated around the kth cycle, as given by algcurves[homology], is Kronecker delta (j, k).

Note: The Abel map will almost always be computed along with other objects associated with some polynomial F, such as the Riemann matrix. It is imperative that the order of the differential be the same for each of the objects, and at each stage of the calculation. As no order is imposed by algcurves[differentials], make sure to compute AbelMap and, for instance algcurves[periodmatrix], without a restart (or quit) in between.

This command is based on code written by Bernard Deconinck, Michael A. Nivala, and Matthew S. Patterson.

$\mathrm{with}\left(\mathrm{algcurves}\&comma;\mathrm{AbelMap}\&comma;\mathrm{genus}\&comma;\mathrm{puiseux}\right)$

$\left[\mathrm{AbelMap}\&comma;\mathrm{genus}\&comma;\mathrm{puiseux}\right]$

$f≔y^2-\left(x^2-1\right)\left(x^2-4\right)\left(x^2-9\right)\left(x^2-16\right)$

$f≔y^2-\left(x^2-1\right)\left(x^2-4\right)\left(x^2-9\right)\left(x^2-16\right)$

$\mathrm{genus}\left(f\&comma;x\&comma;y\right)$

$3$

$\mathrm{puiseux}\left(f\&comma;x=1\&comma;y\&comma;0\&comma;t\right)$

$\left\{\left[x=-720t^2+1\&comma;y=-720t\right]\right\}$

$\mathrm{puiseux}\left(f\&comma;x=4\&comma;y\&comma;0\&comma;t\right)$

$\left\{\left[x=10080t^2+4\&comma;y=10080t\right]\right\}$

$\mathrm{P\_0},P≔\mathrm{op}\left(\right),\mathrm{op}\left(\right)$

$\mathrm{P\_0},P≔\left[x=-720t^2+1\&comma;y=-720t\right],\left[x=10080t^2+4\&comma;y=10080t\right]$

$A≔\mathrm{AbelMap}\left(f\&comma;x\&comma;y\&comma;P\&comma;\mathrm{P\_0}\&comma;t\&comma;7\right)$

$A≔\left[\mathrm{-0.5086732390}-1.395818333\mathrm{I}\&comma;0.5158465233+0.3733240360\mathrm{I}\&comma;0.00716997551-0.3585270829\mathrm{I}\right]$