Let $f=f(t),g=g(t) \in k[t]$ be two even polynomials, namely, each is a sum of even degrees monomials (including degree zero). $k$ is a field of characteristic zero.
Assume that we know that there exist another two even polynomials $A=A(t), B=B(t) \in k[t]$, such that $Af+Bg=1$.
Is there something interesting that we can say about the exact form of $f$ and $g$ (involving their degrees, perhaps)?
Example: I have taken an arbitrary monic even $f$ of degree $4$ and an arbitrary monic even $g$ of degree $6$, and tried to solve $Af+Bg=1$, namely, to find the coefficients of $A$ and $B$ in terms of the coefficients of $f$ and $g$. Actually, I have taken $A$ and $B$ of small given appropriate degrees. Although we get linear equations, which are usually easy to handle, I am not sure how this helps in characterizing our 'original' $f$ and $g$ (their coefficients).
Remark: From $Af+Bg=1$ we get that there exists $h=h(t) \in k[t]$ such that $th= A'f+Af'+B'g+Bg'=0$, but I do not see how this helps.
Thank you very much!