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Given a matrix valued function $f_t:\mathbb R^{N\times N} \mapsto \mathbb R^{N\times N}$. For $f_t(A)=B$ both $A$ and $B$ are symmetric. Which properties can be assigned to $f_t$, when the following division $$ \frac{\det(f_t(A))}{(1-t^2)^{N/2}} $$ will again result in a polynomial. Should a polynomial $f_t(\cdot)$ have some restrictions on the coefficients?

To give an explicit example (see here): Is $\displaystyle \frac{\det((1+2t^2)I - At)}{(1-t^2)^{N/2}}$ expressible as polynomial?

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draks ...
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    I find the question confusing: what is the relationship between $t$ and the numerator of your fraction? Maybe you meant that $f \in \mathcal M _N (\Bbb R) [[t]]$, i.e. is a formal series in $t$ with matrix coefficients? – Alex M. Nov 08 '15 at 16:36
  • @AlexM. $t$ is a parameter of $f()$. I edited the question... – draks ... Nov 08 '15 at 17:36

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