This paper counts how often a natural family of symmetric integer polynomials fails to have the largest possible Galois group. The family is defined by a simple symmetry: a polynomial f of degree 2n satisfies f(x) = x^{2n} b^{-n} f(b/x) for a fixed nonzero integer b. The roots of such a polynomial always come in pairs, and the typical Galois group is the hyperoctahedral group — the group that permutes those pairs and swaps the two elements inside each pair. The authors show that the number of exceptional polynomials (those whose Galois group is not the full hyperoctahedral group) follows an explicit asymptotic law as the coefficient size grows.

To make a fair count they use a height that respects the symmetry. Instead of the usual maximum coefficient size, the height is a Euclidean norm that weights coefficients according to the symmetry: essentially a sum of |b|^i times the square of the i-th coefficient. This choice makes the set of polynomials of bounded height into a simple ellipsoid in coefficient space, and it matches the pairing of roots. Important special cases are b = 1, which gives ordinary reciprocal (palindromic) polynomials, and b = -1, which gives skew-reciprocal polynomials. Such polynomials arise as characteristic polynomials of matrices from symplectic groups and from the symplectic similitude group, so the counting question has links to questions about possible characteristic polynomials in those matrix groups.

The main theorem, valid for fixed b ≠ 0 and integers n ≥ 4, is an asymptotic count of exceptional polynomials of height at most H. For the nonmonic case the number equals an explicit constant (depending on n and b) times H^n log H, plus an error of order H^n. For monic polynomials there is a parallel formula with H^{n-1} log H and error H^{n-1}. The authors identify the precise dependence of the leading constant on b. They also show that most of the exceptional polynomials come from a single index-2 subgroup (called G1 in the paper); all other possible Galois subgroups contribute only lower-order counts.