Cyclotomic type criterion for unimodular matrices

Question

Here unimodular usually means $A \in GL(n,\mathbb{Z})$ (but if you like you can also assume $A \in SL(n, \mathbb{Z})$.

In an article I read the following (the problem comes from representation theory of quiver algebras):

A unimodular $n \times n$ matrix $A$ has all eigenvalues on the unit circle (this is called cyclotomic) if and only if we have $|Tr(A^k)| \leq n$ for all $k=0,...,n$. (maybe there is a typo in the article and it is meant $|Tr(A)^k| \leq n$ instead?)

Usually Im wrong when I find something that looks wrong in an article and since I got noone to ask this in the next weeks I thought about asking here (I can delete the thread in case I made a stupid thinking error,so please do not answer in case I just have a thinking error. ).

I think I found a counterexample with GAP (I post the GAP input/output so you can check it if you like):

Let $A$ be the following unimodular matrix (this is the coxeter matrix of the Nakayama algebra with Kupisch series [5,5,5,5,5,5,4,3,2,1]):

[ [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ], [ -1, 0, 0, 0, 0, 0, -1, -1, -1, -1 ], [ 0, -1, 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, -1, 0, 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, -1, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, -1, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 0, -1, -1, -1, -1, -1 ] ]

Then we have W:=[];for i in [0..10] do Append(W,[AbsoluteValue(Trace(g)^i)]);od;W;

[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]

and

W:=[];;for i in [0..10] do Append(W,[AbsoluteValue(Trace(g^i))]);od;W;

[ 10, 1, 1, 2, 1, 4, 4, 6, 1, 2, 6 ] .

So by the above theorem, $A$ should be cyclotomic. But GAP gives me for the characteristic polynomial x_1^10+x_1^9-x_1^7-x_1^6-x_1^5-x_1^4-x_1^3+x_1+1 and wolfram alpha says it has a zero 1.17628... which does not lie on the unit circle: https://www.wolframalpha.com/input/?i=x_1%5E10%2Bx_1%5E9-x_1%5E7-x_1%5E6-x_1%5E5-x_1%5E4-x_1%5E3%2Bx_1%2B1

Question: In case there is really something wrong, can the criterion be fixed by using $|Tr(A^k)| \leq n$ for $i=0,1,...,f(n)$ instead (so we replace $n$ by a polynomial function $f(n)$. Maybe $f(n)=n^2$ already works (it does in this example)).

It might be interesting to see what a "minimal" f(n) might look like (even if when it is not polynomial). For example how does $f(n)$ start for $n=2,3,...$ (in case it exists). Of course one might also think about this question for various types of matrices, such as $A \in SL(n,K)$ or $GL(n,K)$ , where $K$ is $\mathbb{Z}$, $\mathbb{R}$ or $\mathbb{C}$.

edit: Fun questions:

1) What is the number of cyclotomic 0-1 matrices with determinant 1? It seems to start with 1, 3, 27, 756, see https://oeis.org/A185149.

2) What is the number of cyclotomic 0-1 matrices with determinant 1 or -1? It seems to start with 1, 4, 48, 1536, see https://oeis.org/A011266.

Sadly my computer seems to be willing to calculate this only until n=4...

Answer 1

This doesn't have much to do with matrices, I think. Since being cyclotomic and the trace are conjugacy invariants, we may as well assume that $A$ is in Jordan form, say $A=J+N$ with $J$ diagonal and $N$ upper nilpotent. Then WLOG, we may take $A=J$, since eigenvalues and traces of $J^k$ and $(J+N)^k$ are the same.

Now one direction is easy. If $A$ is cyclotomic with eigenvalues $e_1,\ldots,e_n$, then $$ |\text{Trace}(A^k)| = |e_1^k+\cdots+e_n^k|\le |e_1|^k+\cdots+|e_n|^k=n. $$

For the other direction, let $e_1,\ldots,e_n$ be the eigenvalues of $A$, or for that matter, an arbitrary Galois invariant set of algebraic integers. If we assume that $$ |e_1^k+\cdots+e_n^k|\le C\quad\text{for $k\ge1$,} $$ then by the usual formulas relating elementary symmetric polynomials to elementary power polynomials, the elementary symmetric polys of $e_1,\ldots,e_n$ are bounded by a function of $C$ and $n$, so there are only finitely many possibilities for $e_1,\ldots,e_n$. But any set of powers $e_1^k,\ldots,e_n^k$ has the same property, so for each $i$, the sequence $(e_i^k)_{k\ge1}$ is a finite set, and hence every $e_i$ is a root of unity. (This is all more-or-less Kronecker's theorem, and can undoubtedly be proven more slickly.)