Let $f(X)$ be a polynomial over a finite field, and let its defining function record the multiplicities of its roots. In this paper, we introduce the affine stabilizer group and the affine stabilizer group scheme of the defining function of $f(X)$. These objects characterize the affine symmetries of the roots of $f(X)$ together with their multiplicities. We establish structural descriptions of these affine stabilizers and relate them to the coefficient structure of $f(X)$. In particular, the affine stabilizer group determines quotient structures of the defining function, which yield canonical descriptions of the corresponding semi-invariant polynomial spaces. The resulting coefficient constraints provide explicit restrictions on the possible supports of polynomials with prescribed affine symmetries. The affine stabilizer group scheme further captures infinitesimal symmetries arising from multiple roots and gives refined coefficient constraints beyond those obtained from ordinary affine transformations. We apply this framework to generator polynomials of constacyclic codes over finite fields. The affine stabilizer structure of a generator polynomial induces subcodes defined by semi-invariant conditions and provides constraints on their dimensions and Hamming distances. In the special case of multiplicative symmetries, the construction specializes to the support-primitive decomposition of constacyclic codes.

翻译：暂无翻译