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


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Paradoxe de Cramer — En mathématiques, et plus précisément en géométrie algébrique, le paradoxe de Cramer (nommé d après Gabriel Cramer, mais qui avait déjà été remarqué par Maclaurin) affirme que le nombre de points d intersection de deux courbes de haut degré peut… …   Wikipédia en Français

  • Gabriel Cramer — Infobox Scientist name = Gabriel Cramer image width = 300px caption = Gabriel Cramer (1704 1752). Portrait by an unknown artist. birth date = July 31, 1704 birth place = Geneva, Switzerland death date = January 4, 1752 death place = Bagnols sur… …   Wikipedia

  • St. Petersburg paradox — In economics, the St. Petersburg paradox is a paradox related to probability theory and decision theory. It is based on a particular (theoretical) lottery game (sometimes called St. Petersburg Lottery ) that leads to a random variable with… …   Wikipedia

  • List of misnamed theorems — This is a list of misnamed theorems in mathematics. It includes theorems (and lemmas, corollaries, conjectures, laws, and perhaps even the odd object) that are well known in mathematics, but which are not named for the originator. That is, these… …   Wikipedia

  • Riemann surface — For the Riemann surface of a subring of a field, see Zariski–Riemann space. Riemann surface for the function ƒ(z) = √z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real… …   Wikipedia

  • Plane curve — In mathematics, a plane curve is a curve in a Euclidean plane (cf. space curve). The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. A smooth plane curve is a curve in a …   Wikipedia

  • Georges Henri Halphen — George Henri Halphen (30 October 1844, Rouen – 23 May 1889, Versailles) was a French mathematician. He did his studies at École Polytechnique (X 1862). He was known for his work in geometry, particularly in enumerative geometry and the… …   Wikipedia

  • Moduli of algebraic curves — In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending on… …   Wikipedia

  • De Franchis theorem — In mathematics, the de Franchis theorem is one of a number of closely related statements applying to compact Riemann surfaces, or, more generally, algebraic curves, X and Y, in the case of genus g > 1. The simplest is that the automorphism… …   Wikipedia

  • Genus–degree formula — In classical algebraic geometry, the genus–degree formula relates the degree d of a non singular plane curve with its arithmetic genus g via the formula: A singularity of order r decreases the genus by .[1] Proofs The proof follows immediately… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”