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In [[mathematics]], a '''homogeneous function''' is a [[function of several variables]] such that the following holds: If each of the function's arguments is multiplied by the same [[scalar (mathematics)|scalar]], then the function's value is multiplied by some power of this scalar; the power is called the '''degree of homogeneity''', or simply the ''degree''. That is, if {{mvar|k}} is an integer, a function {{mvar|f}} of {{mvar|n}} variables is homogeneous of degree {{mvar|k}} if
:<math>f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n)</math>
for every <math>x_1, \ldots, x_n,</math> and <math>s\ne 0.</math> This is also referred to a ''{{mvar|k}}th-degree'' or ''{{mvar|k}}th-order'' homogeneous function.
For example, a [[homogeneous polynomial]] of degree {{mvar|k}} defines a homogeneous function of degree {{mvar|k}}.
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There are two commonly used definitions. The general one works for vector spaces over arbitrary [[field (mathematics)|fields]], and is restricted to degrees of homogeneity that are [[integer]]s.
The second one supposes to work over the field of [[real number]]s, or, more generally, over an [[ordered field]]. This definition restricts to positive values the scaling factor that occurs in the definition, and is therefore called ''positive homogeneity'', the qualificative ''positive'' being often omitted when there is no risk of confusion. Positive homogeneity leads to
The restriction of the scaling factor to real positive values allows also considering homogeneous functions whose degree of homogeneity is any real number.
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More generally, every [[norm (mathematics)|norm]] and [[seminorm]] is a positively homogeneous function of degree {{math|1}} which is not a homogeneous function. As for the absolute value, if the norm or semi-norm is defined on a vector space over the complex numbers, this vector space has to be considered as vector space over the real number for applying the definition of a positively homogeneous function.
===Linear
Any [[linear map]] <math>f : V \to W</math> between [[vector space]]s over a [[field (mathematics)|field]] {{mvar|F}} is homogeneous of degree 1, by the definition of linearity:
<math display="block">f(\alpha \mathbf{v}) = \alpha f(\mathbf{v})</math>
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The converse is proved by integrating a simple [[differential equation]].
Let <math>\mathbf{x}</math> be in the interior of the ___domain of {{mvar|f}}. For {{mvar|s}} sufficiently close
<math display="inline"> g(s) = f(s \mathbf{x})</math> is well defined. The partial differential equation implies that
<math display=block>
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