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D.13.6.8 tropicalJInvariant
Procedure from library tropical.lib (see tropical_lib).
- Usage:
- tropicalJInvariant(f[,#]); f poly or list, # optional list
- Assume:
- f is list of linear polynomials of the form ax+by+c with integers
a, b and a rational number c representing a tropical Laurent
polynomial defining a tropical plane curve;
alternatively f can be a polynomial in Q(t)[x,y] defining a
tropical plane curve via the valuation map;
the basering must have a global monomial ordering, two variables
and up to one parameter!
- Return:
- number, if the graph underlying the tropical curve has precisely
one loop then its weighted lattice length is returned,
otherwise the result will be -1
- Note:
- - if the tropical curve is elliptic and its embedded graph has
precisely one loop, then the weighted lattice length of
the loop is its tropical j-invariant
- the procedure checks if the embedded graph of the tropical
curve has genus one, but it does NOT check if the loop can
be resolved, so that the curve is not a proper tropical
elliptic curve
- if the embedded graph of a tropical elliptic curve has more
than one loop, then all but one can be resolved, but this is
not observed by this procedure, so it will not compute
the j-invariant
- if # is empty, then the tropical curve is computed w.r.t. minimum,
if #[1] is the string 'max', then it is computed w.r.t. maximum
- the tropicalJInvariant of a plane tropical cubic is the
'cycle length' of the cubic as introduced in the paper:
Eric Katz, Hannah Markwig, Thomas Markwig: The j-invariant
of a cubic tropical plane curve.
Example:
| | LIB "tropical.lib";
ring r=(0,t),(x,y),dp;
// tropcialJInvariant computes the tropical j-invariant of an elliptic curve
tropicalJInvariant(t*(x3+y3+1)+1/t*(x2+y2+x+y+x2y+xy2)+1/t2*xy);
==> // ** int division with `/`: use `div` instead in line >> genus=-genus/2\
; // we have counted each bounded edge twice<<
==> 6
// the Newton polygon need not be the standard simplex
tropicalJInvariant(x+y+x2y+xy2+1/t*xy);
==> // ** int division with `/`: use `div` instead in line >> genus=-genus/2\
; // we have counted each bounded edge twice<<
==> 8
// the curve can have arbitrary degree
tropicalJInvariant(t*(x7+y7+1)+1/t*(x4+y4+x2+y2+x3y+xy3)+1/t7*x2y2);
==> // ** int division with `/`: use `div` instead in line >> genus=-genus/2\
; // we have counted each bounded edge twice<<
==> 14/3
// the procedure does not realise, if the embedded graph of the tropical
// curve has a loop that can be resolved
tropicalJInvariant(1+x+y+xy+tx2y+txy2);
==> // ** int division with `/`: use `div` instead in line >> genus=-genus/2\
; // we have counted each bounded edge twice<<
==> 4
// but it does realise, if the curve has no loop at all ...
tropicalJInvariant(x+y+1);
==> // ** int division with `/`: use `div` instead in line >> genus=-genus/2\
; // we have counted each bounded edge twice<<
==> The embedded graph of the curve has not genus one.
==> -1
// or if the embedded graph has more than one loop - even if only one
// cannot be resolved
tropicalJInvariant(1+x+y+xy+tx2y+txy2+t3x5+t3y5+tx2y2+t2xy4+t2yx4);
==> // ** int division with `/`: use `div` instead in line >> genus=-genus/2\
; // we have counted each bounded edge twice<<
==> The embedded graph of the curve has not genus one.
==> -1
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