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In mathematics, the degree of an affine or projective variety of dimension n is the number of intersection points of the variety with n hyperplanes in general position. [1] For an algebraic set, the intersection points must be counted with their intersection multiplicity, because of the possibility of multiple components.
Hypersurfaces have some specific properties that are not shared with other algebraic varieties. One of the main such properties is Hilbert's Nullstellensatz, which asserts that a hypersurface contains a given algebraic set if and only if the defining polynomial of the hypersurface has a power that belongs to the ideal generated by the defining polynomials of the algebraic set.
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 ...
Implicit means that the equation is not expressed as a solution for either x in terms of y or vice versa. If (,) is a polynomial in two variables, the corresponding curve is called an algebraic curve, and specific methods are available for studying it.
A surface is doubly ruled if through every one of its points there are two distinct lines that lie on the surface. The hyperbolic paraboloid and the hyperboloid of one sheet are doubly ruled surfaces. The plane is the only surface which contains at least three distinct lines through each of its points (Fuchs & Tabachnikov 2007).
For solving the cubic equation x 3 + m 2 x = n where n > 0, Omar Khayyám constructed the parabola y = x 2 /m, the circle that has as a diameter the line segment [0, n/m 2] on the positive x-axis, and a vertical line through the point where the circle and the parabola intersect above the x-axis.