page_title: Cramer's paradox
hierarchical_section_title: Cramer's paradox
caption_attribution_description: Two cubic curves with nine intersection points
context_page_description: In mathematics, Cramer's paradox or the Cramer–Euler paradox 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.
  This paradox is the result of a naive understanding or a misapplication of two theorems:
  Bézout's theorem.
  Cramer's theorem.
  Observe that for all n ≥ 3, n² ≥ n/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 / 2 points are not enough to determine a curve uniquely.
context_section_description: In mathematics, Cramer's paradox or the Cramer–Euler paradox 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.
  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.
caption: Cubic curves intersecting in 9 points
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