A Kronecker polynomial is a polynomial such that

  1. Its coefficients are integers,
  2. Its roots are on the unit circle or $0$,
  3. Its leading coefficient is $1$.

From here, there are finitely many Kronecker polynomials of a certain degree, and they are of the form

$$f(x)=x^k \prod_j \Phi_j(x)$$

Where $\Phi_n(x)$ is the $n$-th cyclotomic polynomial, ie. Cyclotomic[n,x].

I want to input $N$ and get the set of Kronecker polynomials of degree $N$. To do this, I think I would do the following:

  1. Set up a table for i=1,2...N-1, which will be the power of the $x^k$ factor.
  2. Find all possible unordered $(l_1,l_2 \dots l_r)$ such that $\deg(\prod\Phi_{l_i}) = N-i$.
  3. Compute the polynomial $f(x)$ corresponding to $k$ and $k$'s corresponding $(l_1,l_2 \dots l_r$).
  4. Merge the lists.

However, I have no clue how to achieve step 2! Any help would be appreciated.



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