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Given a homogeneous polynomial of degree with real coefficients that takes only positive values, one gets a positively homogeneous function of degree / by raising it to the power /. So for example, the following function is positively homogeneous of degree 1 but not homogeneous: ( x 2 + y 2 + z 2 ) 1 2 . {\displaystyle \left(x^{2}+y^{2}+z^{2 ...
A linear differential equation is homogeneous if it is a homogeneous linear equation in the unknown function and its derivatives. It follows that, if φ(x) is a solution, so is cφ(x), for any (non-zero) constant c. In order for this condition to hold, each nonzero term of the linear differential equation must depend on the unknown function or ...
A C k field, more generally, is one for which any homogeneous polynomial of degree d in N variables has a non-trivial zero, provided d k < N, for k ≥ 1. [11] The condition was first introduced and studied by Lang. [10] If a field is C i then so is a finite extension. [11] [12] The C 0 fields are precisely the algebraically closed fields. [13 ...
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. [1] For example, x 5 + 2 x 3 y 2 + 9 x y 4 {\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}} is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5.
The resultant of two polynomials depending on a variable x and other variables is a polynomial in the other variables that is in the ideal generated by the two polynomials, and has the following properties: if one of the polynomials is monic in x, every zero (in the other variables) of the resultant may be extended into a common zero of the two ...
The polynomial ring = [, …,] is graded by degree: it is a direct sum of consisting of homogeneous polynomials of degree i. Let S be the set of all nonzero homogeneous elements in a graded integral domain R .
When one changes the coordinates to put the singular point at the origin, the equation of the tangents at the singular point is thus the nonzero homogeneous part of the lowest degree of the polynomial, and the multiplicity of the singular point is the degree of this homogeneous part.
A complete classification of homogeneous distributions in one dimension is possible. The homogeneous distributions on R \ {0} are given by various power functions.In addition to the power functions, homogeneous distributions on R include the Dirac delta function and its derivatives.