Cramer's paradox
Encyclopedia
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.
Similarly two conics intersect at 4 points, and 5 points are needed to define a conic.
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.
Gabriel Cramer
Gabriel Cramer was a Swiss mathematician, born in Geneva. He showed promise in mathematics from an early age. At 18 he received his doctorate and at 20 he was co-chair of mathematics.In 1728 he proposed a solution to the St...
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
Bézout's theorem
Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees...
(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.
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.