EQUIVARIANT HYERS–ULAM STABILITY OF QUADRATIC FUNCTIONAL EQUATIONS ON ABELIAN GROUPS AND POLYNOMIAL MODULES

Abstract

The classical dyadic construction gives a quadratic limit for an approximately quadratic map on an Abelian group under summable control. We prove that endomorphism-intertwining identities pass to this limit when the rescaled intertwining defects vanish. For \(M=(\mathbb{Z}[X]\slash (X^n+1))^k\), coefficient norms give explicit multiplier criteria. Relations between multiplier words force operator identities on the span \(W=\operatorname{span}_{\mathbb F}Q(M)\). This span is finite-dimensional, and \(X^n=-1\) implies \(U^n|_W=I_W\). Expected defects are treated in \(L^1(\Omega;Y)\). McDiarmid'sinequality gives bounds at fixed dyadic stages on finite grids, not a global pathwise stability theorem.