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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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