#math/linear-algebra #math/pde #flashcards/math A function $f:D\subseteq\mathbb{R}^n\to\mathbb{R}$ is **homogeneous of degree $k$** if %%::%% $ f(\lambda x)=\lambda^k f(x) $ for every $x\in D$ and every scalar $\lambda$ for which $\lambda x\in D$ and $\lambda^k$ is defined. <!--SR:!fsrs,2026-09-25T16:14:22.998Z,0,2.3065,2.11810397,1,1,0,1,2026-09-25T16:04:22.998Z--> For [[2A2 - Polynomials|polynomials]], a polynomial is homogeneous when every monomial has the same total degree. For example, $ f(x,y)=x^3+x^2y+xy^2+y^3 $ is homogeneous of degree $3$, because $ f(\lambda x,\lambda y) =(\lambda x)^3+(\lambda x)^2(\lambda y)+(\lambda x)(\lambda y)^2+(\lambda y)^3 =\lambda^3f(x,y). $