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D.5.1.5 rationalPointConic
Procedure from library paraplanecurves.lib (see paraplanecurves_lib).
- Usage:
- rationalPointConic(p); p poly
- Assume:
- assumes that p is an irreducible quadratic polynomial in the first
three ring variables;
ground field is expected to be Q.
- Return:
- The method finds a point on the given conic. There are two
possibilities:
1) There is a rational point on the curve.
2) There is no rational point on the curve.
In the second case, the method creates a modification of the current
basering which is a polynomial ring over some quadratic field
extension Q(a) of Q. Apart from the replacement of Q by Q(a), the
new polynomial ring, R say, is the same as the original basering.
(In the first case, R is identical with the basering.)
In both cases, the method will then define a (1x3) matrix named
'point' which lives in R and which contains the coordinates of the
desired point on q.
Finally, the method returns the ring R (which will in the 1st case
be the original base ring).
Example:
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