Search results
Results from the WOW.Com Content Network
A complete intersection has a multidegree, written as the tuple (properly though a multiset) of the degrees of defining hypersurfaces. For example, taking quadrics in P 3 again, (2,2) is the multidegree of the complete intersection of two of them, which when they are in general position is an elliptic curve.
A smooth real cubic surface is rational over R if and only if its space of real points is connected, hence in the first four of the previous five cases. [ 13 ] The average number of real lines on X is 6 2 − 3 {\displaystyle 6{\sqrt {2}}-3} [ 14 ] when the defining polynomial for X is sampled at random from the Gaussian ensemble induced by the ...
More concretely, because the vector space of homogeneous polynomials P(x, y, z) of degree three in three variables x, y, z has dimension 10, the system of cubic curves passing through eight (different) points is parametrized by a vector space of dimension ≥ 2 (the vanishing of the polynomial at one point imposes a single linear condition).
The twisted cubic has the following properties: It is the set-theoretic complete intersection of and () (), but not a scheme-theoretic or ideal-theoretic complete intersection; meaning to say that the ideal of the variety cannot be generated by only 2 polynomials; a minimum of 3 are needed.
The 2nd Brocard cubic is the locus of a point X for which the pole of the line XX* in the circumconic through X and X* lies on the line of the circumcenter and the symmedian point (i.e., the Brocard axis). The cubic passes through the centroid, symmedian point, both Fermat points, both isodynamic points, the Parry point, other triangle centers ...
A cubic function with real coefficients has either one or three real roots (which may not be distinct); [1] all odd-degree polynomials with real coefficients have at least one real root. The graph of a cubic function always has a single inflection point. It may have two critical points, a local minimum and a local maximum. Otherwise, a cubic ...
The twisted cubic is a projective algebraic variety.. Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics.Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers.
A general straight-line thread connects the two points (0, k−t) and (t, 0), where k is an arbitrary scaling constant, and the family of lines is generated by varying the parameter t. From simple geometry, the equation of this straight line is y = −(k − t)x/t + k − t. Rearranging and casting in the form F(x,y,t) = 0 gives: