There are two questions. The first one is motivated by my failure in computing the second.
1) Suppose that $A : V \rightarrow V$ is a square $n \times n$ matrix defined over a field $k$ such that $$Ann_{k[x]} (V) = (p (x)^d)$$ for $d > 0$, i.e., $p(x)^d$ is the minimal polynomial and $p (x) \in k[x]$ is irreducible.
Let $$dim_k (Ann_{V} (p(A)^{k})) = n_k$$ be given for all values $k \leq d$
By fixing $n$ and $p(x)^d$. Is it true that for every sequence $(n_k)_k$, there exists $A$ satisfying these dimension constraints? Furthermore, is there a criterion for unicity if only some of the $n_k$'s are specified?
2) Let $n = 748$, $n_1 = 12$, $n_2 = 20$, $n_{89} = 360$, $n_{90} = 368$, and $n_{91} = 372$. What's the rational canonical form of $A$? What's the value of $d$?
Let me comment how I would solve the second. $12$ blocks have size at least $4$ (so there are $12$ blocks), $8$ have size at least $8$, $8$ have size at least $360 = 89 . 4$ and $4$ blocks have size at least $364 = 91. 4$. Now, we have to solve $$q_1 + … + q_{12} = 748$$ using the above inequalities, where $q_i$ is size of the $i$-th block. Unfortunately I don't know how to solve it nor I know if there's enough degrees of freedom to find a unique canonical form (although the original question suggest that there's only one possibility). This motivates the first question.
Thanks in advance.
EDIT
Question 2 is easy to solve actually. I was probably very sleepy the day I asked. There are actually $3$ blocks of size $4$. I forgot to divide by $4$. And 1 block of size $8$, by taking the difference $12 - (20 -12)$.