Cramer's paradox

In mathematics, Cramer's paradox or the Cramer–Euler paradox[1] is the statement that the number of points of intersection of two higher-order curves in the plane can be greater than the number of arbitrary points that are usually needed to define one such curve. It is named after the Genevan mathematician Gabriel Cramer.

Cubic curves intersecting in 9 points

This paradox is the result of a naive understanding or a misapplication of two theorems:

  • Bézout's theorem (the number of points of intersection of two algebraic curves is equal to the product of their degrees, provided that certain necessary conditions are met).
  • Cramer's theorem (a curve of degree n is determined by n(n + 3)/2 points, again assuming that certain conditions hold).

Observe that for all n ≥ 3, n2 ≥ n(n + 3)/2, so it would naively appear that for degree three or higher there could be enough points shared by each of two curves that those points should determine either of the curves uniquely.

The resolution of the paradox is that in certain degenerate cases n(n + 3) / 2 points are not enough to determine a curve uniquely.

History

The paradox was first published by Colin Maclaurin.[2][3] Cramer and Leonhard Euler corresponded on the paradox in letters of 1744 and 1745 and Euler explained the problem to Cramer.[4] It has become known as Cramer's paradox after featuring in his 1750 book Introduction à l'analyse des lignes courbes algébriques, although Cramer quoted Maclaurin as the source of the statement.[5] At about the same time, Euler published examples showing a cubic curve which was not uniquely defined by 9 points[4][6] and discussed the problem in his book Introductio in analysin infinitorum. The result was publicized by James Stirling and explained by Julius Plücker.[1]

No paradox for lines and nondegenerate conics

For first order curves (that is lines) the paradox does not occur, because n = 1 so n2 = 1 < n(n + 3) / 2 = 2. In general two distinct lines L1 and L2 intersect at a single point P unless the lines are of equal gradient (slope), in which case they do not intersect at all. 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 nondegenerate conics intersect at most at 4 finite points in the real plane, which is fewer than the 32 = 9 given as a maximum by Bézout's theorem, and 5 points are needed to define a nondegenerate conic.

Cramer's example for cubic curves

In a letter to Euler, Cramer pointed out that the cubic curves x3 − x = 0 and y3 − y = 0 intersect in precisely 9 points (each equation represents a set of three parallel lines x = −1, x = 0, x = +1; and y = −1, y = 0, y = +1 respectively). Hence 9 points are not sufficient to uniquely determine a cubic curve in degenerate cases such as these.

Resolution

A bivariate equation of degree n has 1 + n(n + 3) / 2 coefficients, but the set of points described by the equation is preserved if the equation is divided through by one of the coefficients, leaving one coefficient equal to 1 and only n(n + 3) / 2 coefficients to characterize the curve. Given n(n + 3) / 2 points (xi, yi), each of these points can be used to create a separate equation by substituting it into the general polynomial equation of degree n, giving n(n + 3) / 2 equations linear in the n(n + 3) / 2 unknown coefficients. If this system is non-degenerate in the sense of having a non-zero determinant, the unknown coefficients are uniquely determined and hence the polynomial equation and its curve are uniquely determined. But if this determinant is zero, the system is degenerate and the points can be on more than one curve of degree n.

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References
  1. Weisstein, Eric W. "Cramér-Euler Paradox." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Cramer-EulerParadox.html
  2. Maclaurin, Colin (1720). Geometria Organica. London.
  3. Tweedie, Charles (January 1891). "V.—The "Geometria Organica" of Colin Maclaurin: A Historical and Critical Survey". Transactions of the Royal Society of Edinburgh. 36 (1–2): 87–150. Retrieved 28 September 2012.
  4. Struik, D. J. (1969). A Source Book in Mathematics, 1200-1800. Harvard University Press. p. 182. ISBN 0674823559.
  5. Tweedie, Charles (1915). "A Study of the Life and Writings of Colin Maclaurin". The Mathematical Gazette. 8 (119): 133–151. JSTOR 3604693.
  6. Euler, L. "Sur une contradiction apparente dans la doctrine des lignes courbes." Mémoires de l'Académie des Sciences de Berlin 4, 219-233, 1750
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