Analogue of Smith normal form for matrices over $\mathbb Z[t]$

Question

Let $R$ be a principal ideal domain and $A \in M_n R$. It is well known that there exist invertible matrices $Q$ and $S$ and a diagonal matrix $D= {\rm diag}(a_1,\dots,a_n)$ such that

• $a_i \mid a_{i+1}$ for all $1 \leq i \leq n-1$, and
• D=QAS.

The matrix $D$ is called Smith normal form of $A$ and is unique. Obviously, this applies to both $\mathbb Z$ and $\mathbb R[t]$.

Question 1: Is there any analogue for $R= \mathbb Z[t]$? Is there any classification of matrices over $\mathbb Z[t]$ up to equivalence?

A related question is the following:

Question 2: Is there any classification of $n \times n$-matrices over $\mathbb Z$ up to conjugation?

(The relation comes from looking at the matrix $t\cdot 1_n - A$ for $A \in M_n \mathbb Z$. Then, classification of $A$ up to conjugation is the same as classification of $t\cdot 1_n-A \in M_n \mathbb Z[t]$ up to equivalence.)

An obvious first invariant is the characteristic polynomial. Even if the matrix is assumed to be symmetric, it is unclear to me what kind of additional information could be added.

In that respect I know of a theorem of Latimer and MacDuffee which says that if the characteristic polynomial $f$ of $A \in M_n \mathbb Z$ is irreducible, then conjugacy classes of integer matrices with the same characteristic polynomial are in bijection with $\mathbb Z[\alpha]$-ideal classes in ${\mathbb Q}(\alpha)$, where $\alpha$ is a root of $f$. However, this seems to depend on the irreducibility of $f$ and I do not know of an extension to the general case. (This is nicely explained in notes by Keith Konrad.)

Question 3: Are there any other positive results going in the direction of the Theorem of Latimer-MacDuffee? What if the characteristic polynomial is a product of two irreducible polynomials?

Related but maybe easier:

Question 4: Is there some characterization (in terms of the characteristic polynomials + additional invariants) of pairs of matrices in $A,B \in M_n\mathbb Z$, such that $A$ and $B$ are conjugate in $M_n \overline{\mathbb F}_p$ for all primes $p$ and in $M_n \mathbb C$?

And finally

Question 5: Is there some characterization (in terms of the characteristic polynomials + additional invariants) of pairs of matrices in $A,B \in M_n\mathbb Z$, such that $A$ and $B$ are conjugate in $M_n \overline{\mathbb F}_p$ for a fixed prime $p$ and in $M_n \mathbb C$?

(Of course, Question 4 and 5 can be answered by looking at the Jordan decomposition for each of the fields separately. However, the question is, can we do something more conceptual?)

Answer 1

For Q1 the problem is that one invariant of the matrix is the (isomorpism class of the) cokernel and any $\mathbb Z[t]$-module generated by $n$ elements and $n$-relations appears as such an invariant. There simply are too many modules over a $2$-dimensional ring such as $\mathbb Z[t]$.

As for Q2 you cannot really hope for a classifiction even for the conjugacy classes of matrices of finite order. In fact already for matrices of order $p^m$, $p$ prime and $m>2$, the problem is wild which essentially means that any complete classification is hopeless.

Finally (as far as these comments go) for the part of Q3 where the characteristic polynomial is the product of two distinct irreducible polynomials $f$ and $g$: If $M$ is $\mathbb Z^n$ as a module of $\mathbb Z[t]$ through the matrix, then we have a direct sum part $M'\bigoplus M''\subseteq M$, where $M'$ is the annihilator of $f$ and $M''$ of $g$. Hence, $M'$ and $M''$ are given by ideals as in Latimer-McDuffee and can be considered classified. Then $M$ is given by a submodule $\overline M$ of $M'\bigotimes\mathbb Q/\mathbb Z\bigoplus M''\bigotimes\mathbb Q/\mathbb Z$. As $M'$ and $M''$ are the kernels of multiplication by $f$ resp.\ $g$ we also get that $\overline M\cap M'\bigotimes\mathbb Q/\mathbb Z=0$ and the same for $M''\bigotimes\mathbb Q/\mathbb Z$. Hence $\overline M$ is the graph of an isomorphism between submodules $\overline M'\subseteq M'\bigotimes\mathbb Q/\mathbb Z$ and $\overline M''\subseteq M''\bigotimes\mathbb Q/\mathbb Z$. This means that $\overline M'$ and $\overline M''$ are killed by both $f$ and $g$ and as they are relatively prime, the kernel of $g$ on $M'\bigotimes\mathbb Q/\mathbb Z$ (as well as the kernel of $f$ on $M''\bigotimes\mathbb Q/\mathbb Z$) are finite (and usually quite small). Hence one can (in principle) determine the possible $\overline M$ and they have to be considered modulo automorphisms of $M'$ and $M''$ which are given by units in their endomorphism rings (which are overorders of $\mathbb Z[t]/(f)$ resp. $\mathbb Z[t]/(g)$).

This sometimes works very well. For instance this is exactly one way of doing the classification of matrices of order $p$ (where $f=t-1$ and $g=t^{p-1}+\cdots+t+1$). On the other hand I am pretty sure it is as hopeless as the general conjugacy problem for general $f$ and $g$.