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I am trying to formally find the class of functions $f: \mathbb{R}^n \rightarrow \mathbb{R}$ that are not homogeneous of degree 1 but whose $n$ partial derivatives are each homogeneous of degree 0. I realized that any function linear in its arguments with a nonzero constant term (i.e. of the form $f(x_1, x_2,..., x_n)= A+\sum_{i=1}^{n}a_ix_i, \; A \neq 0$) satisfies this rule since constant functions are homogeneous of degree 0 but not of degree 1 (except for the $0$ function which I take is trivially homogeneous of any degree). I suspect that this property would hold if we replaced the summation term in $f$ with any differentiable function that is homogeneous of degree 1 (e.g. $x_1^2/x_2, \; x_2 \neq 0$) but I am not sure how to prove this is a sufficient condition to satisfy the property. Also, is having this form (i.e. nonzero constant + homogeneous function of degree 1) a necessary condition to satisfy the property, and if so how can I prove it?

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  • $\begingroup$ Homogenous means closed. A closed algebra is null to its splitting field. Consider the limit in which a coeffient approaches its variable as an identity element for the group it belongs to. $\endgroup$ – Cppg Mar 26 at 22:39

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