Cramer's theorem (algebraic curves)

In mathematics, Cramer's theorem on algebraic curves gives the necessary and sufficient number of points in the real plane falling on an algebraic curve to uniquely determine the curve in non-degenerate cases. This number is

where n is the degree of the curve. The theorem is due to Gabriel Cramer, who published it in 1750.

For example, a line (of degree 1) is determined by 2 distinct points on it: only one line goes through those two points. Likewise, a non-degenerate conic (polynomial equation in x and y with the sum of their powers in any term not exceeding 2, hence with degree 2) is uniquely determined by 5 points in general position (no three of which are on a straight line).

The intuition of the conic case is this: Suppose the given points fall on, specifically, an ellipse. Then five pieces of information are necessary and sufficient to identify the ellipse—the horizontal location of the ellipse's center, the vertical location of the center, the major axis (the length of the longest chord), the minor axis (the length of the shortest chord through the center, perpendicular to the major axis), and the ellipse's rotational orientation (the extent to which the major axis departs from the horizontal). Five points in general position suffice to provide these five pieces of information, while four points do not.

The number of distinct terms (including those with a zero coefficient) in an n-th degree equation in two variables is (n + 1)(n + 2) / 2. This is because the n-th degree terms are ${\displaystyle x^{n},\,x^{n-1}y^{1},\,\dots ,\,y^{n},}$ numbering n + 1 in total; the (n − 1) degree terms are ${\displaystyle x^{n-1},\,x^{n-2}y^{1},\,\dots ,\,y^{n-1},}$ numbering n in total; and so on through the first degree terms ${\displaystyle x}$ and ${\displaystyle y,}$ numbering 2 in total, and the single zero degree term (the constant). The sum of these is (n + 1) + n + (n – 1) + ... + 2 + 1 = (n + 1)(n + 2) / 2 terms, each with its own coefficient. However, one of these coefficients is redundant in determining the curve, because we can always divide through the polynomial equation by any one of the coefficients, giving an equivalent equation with one coefficient fixed at 1, and thus [(n + 1)(n + 2) / 2] − 1 = n(n + 3) / 2 remaining coefficients.

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