# Conditions on the characteristic polynomial of a matrix with sines and cosines to has integer coefficients

Let $$A\in \mathsf{GL}(4,\mathbb{R})$$ be the following matrix:

$$A=\begin{pmatrix} \cos a&-\sin a&0&0\\ \sin a&\cos a&0&0\\0&0&\cos b&-\sin b\\0&0&\sin b&\cos b \end{pmatrix}$$

Assume that $$A$$ has finite order, i.e. $$a$$ and $$b$$ are rational multiples of $$\pi$$.

The problem is to give necessary conditions on the characteristic polynomial of $$A$$ to make it integer, i.e. $$P_A(\lambda)\in\mathbb{Z}[\lambda]$$

The characteristic polynomial is $$P_A(\lambda)=\lambda^4-2\lambda^3(\cos a+\cos b)+\lambda^2(2+4\cos a\cos b)-2\lambda(\cos a+\cos b)+1$$ (it is a symmetric polynomial).

Now we want that

$$\begin{cases} 2(\cos a+\cos b)\in\mathbb{Z}\\ 4\cos a\cos b\in \mathbb{Z} \end{cases}$$

Question: In this case I've solved the system by hand and found the possible values of $$a$$ and $$b$$ but I wonder if there is a more efficient way to do this, or if there is some deeper theory involved, because I want to generalize to higher values (for example when there are three blocks, $$a$$, $$b$$, $$c$$ and so on). I see that in someway elementary symmetric polynomials appear but I don't know if this helps.

Any comment or suggestion will be appreciated!

• We can reframe the question in terms of polynomials. In particular, it is equivalent to consider the problem of finding all degree-$4$ integer polynomials $p(x)$ for which $p$ has no real roots and $p(x)$ divides $x^n - 1$ for some integer $n$. Equivalently, the problem amounts to finding all degree $4$ cyclotomic polynomials. Commented Jun 20, 2020 at 12:45
• Rather, it amounts to finding all polynomials of degree $4$ that can be built as a product of cyclotomic polynomials of degree $2$ or greater. Commented Jun 20, 2020 at 12:54

We'll say that $$A$$ is block diagonal with $$k$$ blocks of size $$2 \times 2$$ with each $$2 \times 2$$ block in the sine/cosine form you present, and $$A$$ has finite order.
The problem of counting the possible characteristic polynomials of $$A$$ amounts to counting the polynomials $$p(x)$$ of degree $$2k$$ for which $$p(x) \mid x^n - 1$$ for some positive integer $$n$$ and $$p(x)$$ has no real roots. Equivalently, we want the number of degree $$2k$$ polynomials that can be written as a product of cyclotomic polynomials with degree at least $$2$$.
Because the degree the cyclotomic polynomial $$\Phi_n$$ is $$\varphi(n)$$, where $$\varphi$$ denotes the totient function, we can reframe the problem as follows: we want to count the number of multisets (sets with repetition) $$S \subset \{3,4,5,\dots\}$$ that satisfy the condition $$\sum_{j \in S} \varphi(j) = 2k.$$ Because the totient function has lower bounds such as $$\varphi(n) \geq \sqrt{n/2}$$, this problem could be solved by "brute force". For example, it is sufficient to take $$S \subset \{3,4,\dots, 8k^2\}$$, and the number of elements in $$S$$ is at most $$k$$.
This is also made much easier if we happen to know the number of integers satisfying $$\varphi(n) = j$$ for $$1 \leq j \leq 2k$$, as in the list given here or in this OEIS entry.
To address your $$k = 2$$ problem in detail, it suffices to note that there are $$3$$ choices of $$n$$ with $$\varphi(n) = 2$$ (namely $$3,4,6$$), and $$4$$ choices of $$n$$ with $$\varphi(n) = 4$$ (namely $$5,8,10,12$$). Thus, the total number of possible characteristic polynomials for the matrix satisfying the criteria of your problem will simply be $$\binom{3 + 2 - 1}{2} + 4 = 6 + 4 = 10,$$ where we make use of the multiset coefficient here. The corresponding number of matrices is $$3^2 + 4 = 13.$$