Consider the function $$f(X)=(f_1(X),f_2(X)), ~~X=(x_1,x_2)$$ over the $p$-adic number field. So $f$ is a vector valued function from $\mathbb Q_p^2 \to \mathbb Q_p^2$.
How to compute the $p$-adic additive valuation $v(f(X)$ ?
For example, consider $f(X)=\sum c_IX^I$, where $c_I=(a_1,a_2) \in \mathbb Q_p \times \mathbb Q_p$ is $2$-tuple and $I=(i_1,i_2)$ is the index set.
What would be $v(f(X))$ ?
If $m(X)=C_IX^I$ be a monomial of $f(X)$ with $C_I=(p,p)$, then I think $v(m(X))=v((p,p))+ |I|$, where $|I|=i_1+i_2$.
Note $|I| \geq 1$. But what is $v((p,p))$ ?
I think $v(m(X)) \geq 2$ and so $v(f(X)) \geq 2$.
Any guidance please.