On kronecker polynomials

Monic polynomials with integer coefficients having all their roots in the unit disc have been studied by Kronecker; they are called Kronecker polynomials. Let n ≥ 1 be an integer. By a strong Kronecker polynomial, we mean a monic polynomial P(X) ∈ Z[X] of degree n -1 and such that P(X) divides P(X^t) for each t € {1, . . . , n - 1}. We say that P(X) is an absolutely Kronecker polynomial if P(X) divides P(X^t ) for each positive integer t. We describe a canonical form of strong (respectively absolute) Kronecker polynomials. We, also, prove that if n is composite, then each strong Kronecker polynomial with degree n - 1 is absolutely Kronecker. If n is prime, then we prove that each strong Kronecker polynomial P(X) ≠ 1 + X + X² + ... + X^n-1 is absolutely Kronecker.