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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.
In contrast, deciding if a generic quartic polynomial of degree four (or higher even degree) is convex is a NP-hard problem. [3] The first counterexample of a polynomial which is convex but not SOS-convex was constructed by Amir Ali Ahmadi and Pablo Parrilo in 2009. [4] The polynomial is a homogeneous polynomial that is sum-of-squares and given ...
More concretely, an n-ary quadratic form over a field K is a homogeneous polynomial of degree 2 in n variables with coefficients in K: (, …,) = = =,. This formula may be rewritten using matrices: let x be the column vector with components x 1 , ..., x n and A = ( a ij ) be the n × n matrix over K whose entries are the coefficients of q .
An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
The real regular solid harmonics, expressed in Cartesian coordinates, are real-valued homogeneous polynomials of order in x, y, z. The explicit form of these polynomials is of some importance. They appear, for example, in the form of spherical atomic orbitals and real multipole moments. The explicit Cartesian expression of the real regular ...
Furthermore, if the polynomial has a degree 2d greater than two, there are significantly many more non-negative polynomials that cannot be expressed as sums of squares. [4] The following table summarizes in which cases every non-negative homogeneous polynomial (or a polynomial of even degree) can be represented as a sum of squares:
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables (,) = + +,where a, b, c are the coefficients.When the coefficients can be arbitrary complex numbers, most results are not specific to the case of two variables, so they are described in quadratic form.
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 ...