Consider polynomials
\[ f_{1}(x, y)=\sum_{i, j=0}^{\infty} a_{i j} x^{i} y^{j} \quad \text { and } \quad f_{2}(x, y)=\sum_{i, j=0}^{\infty} b_{i j} x^{i} y^{j} \in \mathbb{R}[x, y] \]
$($where $a_{i j}=b_{i j}=0$ for all but finitely many $(i, j) \in \mathbb{N}^{2} ),$ such that $f_{1}(p, q)=f_{2}(p, q)$ for all $(p, q) \in \mathbb{R}^{2}$ satisfying $p^{2}=q^{2}$. Which of the following sentences is true for all such $f_{1}$ and $f_{2}?$
- $a_{00}=b_{00}$, but we may not have $a_{i j}=b_{i j}$ for all $(i, j)$ with $i+j=1$.
- $a_{i j}=b_{i j}$ if $i+j \leq 1$, but we may not have $a_{i j}=b_{i j}$ for all $(i, j)$ with $i+j=2$.
- $a_{i j}=b_{i j}$ if $i+j \leq 2$, but we may not have $a_{i j}=b_{i j}$ for all $(i, j)$ with $i+j=3$.
- $a_{i j}=b_{i j}$ if $i+j \leq 3$, but we may not have $f_{1}=f_{2}$.
