Cramer's theorem (algebraic curves)

${\displaystyle \frac{n(n+3)}{2},}$

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: one and 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.

Derivation of the formula

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.

For example, a fourth degree equation has the general form

${\displaystyle x^4+c_1{x}^{3}y+c_2{x}^2{y}^2+c_{3}x{y}^3+c_4{y}^4+c_5{x}^3+c_6{x}^{2}y+c_{7}x{y}^2+c_8{y}^3+c_9{x}^2+c_{10}xy+c_{11}{y}^2+c_{12}x+c_{13}y+c_{14}=0,}$

with 4(4+3)/2 = 14 coefficients.

Determining an algebraic curve through a set of points consists of determining values for these coefficients in the algebraic equation such that each of the points satisfies the equation. Given n(n + 3) / 2 points (x_i, y_i), 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. More than this number of points would be redundant, and fewer would be insufficient to solve the system of equations uniquely for the coefficients.