A function with the property that multiplying all arguments by a constant changes the value of the function by a monotonic function of that constant: F(kV)=g(k)F(V), where F(·) is the homogeneous function, V is a vector of arguments, k>0 is any constant, and g(·) is some strictly increasing positive function.

Source: Deardorff's Glossary of International Economics

If g(k) = k^X then F(·) is said to be homogeneous of degree X.

f(x,y) = 3x² + xy + 2y² is homogeneous of degree 2.

f(x,y,z) = x³ + 3x²y + 3xy² + y³ + 9xyz is homogeneous of degree 3.