Let $k$ be a ring and let $M$ be a finite free $k$-module.
I am interested in two power series in $k[[t]]$ produced by any endomorphism of $M$. One is $(\exp \circ \ tr ∘ \log) φ$, where we use $M[[t]]$ to store $tr ∘ \log φ$. I am also supposing $k$ has $1/n!$, and I am mostly interested in ℂ.
I kind of wanted to get to the point here though:
this should be related to $Π_{(0 ≤ n ≤ N)} (-1)ⁿdet(1 + t(Λⁿφ))$ in an equation, maybe with a tweak, but something like this:
$Π_{(0 ≤ n ≤ N)} (-1)ⁿdet(1 + t(Λⁿφ)) = (exp ∘ tr ∘ \log) φ : k[[t]]$
I like this equation because it expresses the characteristic polynomial of $φ$ using an entropy-like expression. On one side eigenvalues occur where on the other side they weren't motivated.
Can anyone show the above equation? Can you show it without calculus? Can you show it with only basic facts about generating functions such as $1/(1-x) = \sum x^n$? On the other hand, perhaps the second derivative is necessary since the convexity of log is very important as far as entropy is concerned.