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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.
For example, + + is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions. 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 /.
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 .
The complete homogeneous symmetric polynomials are characterized by the following identity of formal power series in t: = (, …,) = = = = = (this is called the generating function, or generating series, for the complete homogeneous symmetric polynomials).
Pages in category "Homogeneous polynomials" The following 18 pages are in this category, out of 18 total. This list may not reflect recent changes. ...
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 the case of polynomials in more than one indeterminate, a polynomial is called homogeneous of degree n if all of its non-zero terms have degree n. The zero polynomial is homogeneous, and, as a homogeneous polynomial, its degree is undefined. [c] For example, x 3 y 2 + 7x 2 y 3 − 3x 5 is homogeneous of degree 5. For more details, see ...
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 ...