Isomorphic Invariant Orders Without Hermite Equivalence

Bhargava, Evertse, Győry, Remete and Swaminathan asked whether, in every degree n ≥ 4, there are integral polynomials with isomorphic invariant orders that are not Hermite equivalent; they answered the question for n = 4. This paper answers it for every even n and for every n with n − 2 divisible by 3, 5 or 7, with explicit pairs. In degree six one polynomial of the pair can be chosen monic. LaTeX source, figures, PDF and a verification script.