Explicitly, it is used to define the degree of a polynomial and the notion of homogeneous polynomial, as well as for graded monomial orderings used in formulating and computing Taylor series in several variables.
32.
The flag variety may be identified with the complex projective line with homogeneous coordinates and the space of the global sections of the line bundle is identified with the space of homogeneous polynomials of degree on.
33.
Let P, Q be two homogeneous polynomials respectively of degree d ^ \ circ ( P ) and d ^ \ circ ( Q ) with N variables, then, the following inequality holds:
34.
Let P, Q be two homogeneous polynomials respectively of degree d ^ \ circ ( P ) and d ^ \ circ ( Q ) with N variables, then, the following inequality holds:
35.
Any nonzero polynomial may be decomposed, in a unique way, as a sum of homogeneous polynomials of different degrees, which are called the "'homogeneous components "'of the polynomial.
36.
Ellipsoids are examples of algebraic varieties; and so, for general rank, symmetric tensors, in the guise of homogeneous polynomials, are used to define projective varieties, and are often studied as such.
37.
:If no seven points out of are co-conic, then the vector space of cubic homogeneous polynomials that vanish on ( the affine cones of ) ( with multiplicity for double points ) has dimension two.
38.
Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree \ ell in ( x, y, z ) that obey Laplace's equation.
39.
A useful property of the degree reverse lexicographical order is that a homogeneous polynomial is a multiple of the least indeterminate if and only if its leading monomial ( its greater monomial ) is a multiple of this least indeterminate.
40.
For each set " S " of homogeneous polynomials, define the zero-locus of " S " to be the set of points in on which the functions in " S " vanish:
