“Rational Point”的概念、定义、习题解题方法及翻译-数学辞典

单词

Rational Point

释义

Rational Point

A -rational point is a point on an Algebraic Curve, where and are in a Field .

The rational point may also be a Point at Infinity. For example, take the Elliptic Curve

and homogenize it by introducing a third variable

so that each term has degree 3 asfollows:

Now, find the points at infinity by setting

, obtaining

Solving gives

equal to any value, and (by definition)

. Despite freedom in the choice of

, there is only asingle Point at Infinity because the two triples (

), (

) are considered to beequivalent (or identified) only if one is a scalar multiple of the other. Here, (0, 0, 0) is not considered to be a validpoint. The triples (

, 1) correspond to the ordinary points (

), and the triples (

, 0) correspond tothe Points at Infinity, usually called the Line at Infinity.

The rational points on Elliptic Curves over the Galois Field GF() are 5, 7, 9, 10, 13, 14,16, ... (Sloane's A005523).

See also Elliptic Curve, Line at Infinity, Point at Infinity

References

Sloane, N. J. A. SequenceA005523/M3757in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

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