Most undergraduate books on Thermodynamics offer intuitive definitions for intensive and extensive thermodynamic variables. Authors assert, for example, that the former is independent of the system's size while the latter is not. Other authors argue splitting the system into several subsystems and investigating if the quantity of interest changes or remains the same in those. While all approaches make physical sense, I haven't seen someone define those two concepts in a rigorous manner (even now that I am in university).

In the question [Understanding "natural variables" of the thermodynamic potentials using the example of the ideal gas](https://physics.stackexchange.com/questions/307026/understanding-natural-variables-of-the-thermodynamic-potentials-using-the-exam), one person argued in the answers that an extensive quantity, say $A$ which is dependent from other thermodynamic variables $X\_1, X\_2,..., X\_n$ is called extensive *iff* $$ A(\lambda X\_1,\lambda X\_2,...,\lambda X\_n)=\lambda A(X\_1,X\_2,...,X\_n) \ \forall \lambda \in \mathbb R \ \ \tag{1}$$
In other words, a quantity $A $ is intensive if it is described by a *first-order homogeneous* multivariable function. On the other hand, a quantity $B$ is intensive iff
$$ B(\lambda X\_1,\lambda X\_2,..., \lambda X\_n)=B(X\_1,X\_2,...,X\_n) \ \forall \lambda \in \mathbb R \tag{2}$$
which makes $B$ a $0$ degree *homogeneous* function.

While the preceding definitions make sense, I don't get why an extensive variable should be a homogeneous function of first degree. For instance, an ideal gas has a potential energy function
$$ U(N,T) := \frac{3}{2}Nk\_BT,$$
with $N$ being the number of particles and $T$ the temperature of our system.
Obviously, the "test" (1) fails for most choices of $ λ $ (except the trivial cases $0$ and $1$) as it is:

$$ U(λN,λT)= \lambda^2 \* U(N,T)$$
which is a second-order *homogeneous* function.
Should one extend $(1)$ to be:

$$ A(λX\_1,λX\_2,...,λX\_n)=λ^kA(X\_1,X\_2,...,X\_n) \ \forall λ\in \mathbb R \ \ (1)$$
for some $k \in \mathbb N^\* $? and thus defining an extensive quantity as a homogeneous function of $k$ degree. Are there other rigorous definitions out there?