Cramer's paradox

Cramer's paradox

In mathematics, Cramer's paradox (named after Gabriel Cramer but apparently not discovered by him) is the statement that the number of points of intersection of two higher-order curves can be greater than the number of arbitrary points needed to define one such curve.

Cramer's paradox is the result of two theorems: Bézout's theorem (the number of points of intersection of two algebraic curves is equal to the product of their degrees) and a theorem of Cramer (a curve of degree n is determined by n(n + 3)/2 points). Observe that for n ≥ 3, it is the case that n2 is greater than or equal to n(n + 3)/2.

Contents

No paradox for lines and conics

For first order curves (that is lines) the paradox does not occur. In general two lines L1 and L2 intersect at a single point P. A single point is not sufficient to define a line (two are needed); through the point P there pass not only the two given lines but an infinite number of other lines as well.

Similarly two conics intersect at 4 points, and 5 points are needed to define a conic.

The paradox illustrated: cubics and higher curves

By Bézout's theorem two cubics (curves of degree 3) intersect in 9 points. By Cramer's theorem, 9 arbitrary points define a unique cubic. At first thought it may seem that the number of intersection points is too high, defining a unique cubic rather than the two separate cubics that meet there.

Similarly for two curves of degree 4, there will be 16 points of intersection. Through 16 points (assuming they are arbitrarily given) we will usually not be able to draw any quartic curve (14 points suffice), let alone two intersecting quartics.

Resolution

Cramer's proposed resolution turned out to be flawed, but the correct answer was found by Julius Plücker: the points of intersection are not arbitrary; although they are n2 in number, the number of degrees of freedom is d < n2, so that if d of them are given the remaining ones can be determined.

References

External links


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