Ch. Hermite [J. Reine Angew. Math. 53, 182–192 (1857; ERAM 053.1391cj)] attached to a polynomial \[ f(X)=\sum_{j=0}^n f_{n-j}X^j =f_0\prod_{i=1}^n(X-\alpha_i)\in \mathbb Z[X] \] the quadratic form \[ [f](\bar X)=a^{n-1}\prod_{i=1}^n\sum_{j=1}^n\alpha_i^{n-j}X_j, \] where \(\bar X\) is the column vector \((X_1,\dots,X_n)\), and called two polynomials \(f,g\in \mathbb Z[X]\) of the same degree equivalent if with some matrix \(U\in \mathrm{GL}_n(\mathbb Z)\) one has \([g](U\bar X) = \pm[f](\bar X)\).
The authors present in Theorem 1.2 an interpretation of Hermite equivalence in terms of invariant orders \(R_f\) in \(\mathbb Q[X]/(f)\) and show that two monic polynomials \(f,g\) of non-zero discriminant are Hermite equivalent if and only if the associated invariant orders \(R_f,R_g\) are isomorphic. This may fail for non-monic polynomials (such examples are given in Theorem 1.4) but is true under an additional condition. This implies in particular that if the monic polynomials \(f,g\) are \(\mathrm{GL}_2(\mathbb Z)\)-equivalent, then they are also Hermite equivalent.
Theorem 1.3 gives explicit bounds for the number of Hermite equivalence classes of polynomials of given degree and discriminant and shows that every such class has a representative with bounded height. Moreover bounds are given for the number of \(\mathrm{GL}_2(\mathbb Z)\)-equivalence and \(\mathbb Z\)-equivalence classes of polynomials in a Hermite equivalence class.