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The prime ideals of $\mathbb{R}[x]$ are $(0)$ and the maximal ideals generated by the monic irreduciblesFor a nonzero prime $I=(p)$, use the residue classes of $1,x,\dots ,x^{N-1}$, where $N=n\mathrm{deg}p$. Multiplication by $x$ on $\mathbb{R}[x]/(p^n)$ is represented by the companion matrix of the monic polynomial $p^n$. In particular, for $p=x-a$ this may equivalently be written as one size-$n$ Jordan block with eigenvalue $a$. The ideal $(0)$ does not give a finite-dimensional quotient.

For a two-level Fourier recurrence ${\hat{u}}^{n+1}=a(\theta ){\hat{u}}^n+b(\theta ){\hat{u}}^{n-1}$, the amplification matrix isStability requires its powers to be uniformly bounded in the time index and the mesh frequencies. If its two amplification roots have modulus at most one and their separation has a positive mesh-independent lower bound, the eigenvectors $({\xi }_j,1{)}^T$ give a uniformly bounded diagonalization of a matrix, proving stability. A repeated unit-modulus root of this companion matrix instead produces a Jordan block and linear growth in time. Thus checking only the moduli of the roots is insufficient.

For weak quadratic coupling between frequencies one and two, the first oscillator's complex amplitude can obey $2i{A}_T+\alpha A+b\overline{A}=0$ with a constant real second-oscillator amplitude $b$. In real coordinates this is a linear slow system with squared growth rate $(b^2-{\alpha }^2)/4$. Exponential amplification occurs when the modulation amplitude exceeds detuning, $|b|>|\alpha |$. The method of multiple scales is valid over bounded slow time while the physical amplitudes remain in the weak-coupling regime. At equality the slow matrix can have a Jordan block and produce linear growth for some initial phases.

Let the rotation and reflection generators act on polynomials of degree at most two byThen $S^2=I$ and $STS=T^{-1}$. The operator $T$ has one size-three unipotent Jordan block, so this three-dimensional representation cannot split into invariant subspaces of dimension at most two.

The model matrix $B=\left(\begin{array}{cc}0& 1\\ ϵ& d\end{array}\right)$ has a defective double eigenvalue when $d=ϵ=0$. Its characteristic polynomial is ${\lambda }^2-d\lambda -ϵ$, giving the displayed roots. For fixed nonzero $d$ the small root is $-ϵ/d+O({ϵ}^2)$ and the other is $d+ϵ/d+O({ϵ}^2)$. These expansions cease to be ordered at $d=O(\sqrt{ϵ})$, the distinguished limit in which both roots split on a square-root scale. This is the generic leading behavior when a perturbation closes a two-dimensional Jordan block through a nonzero lower-left entry; it need not occur for a perturbation that leaves that entry zero.

For a uniformly bounded two-level Fourier companion matrix $C(\theta )$ with distinct roots $g_1,g_2$ of modulus at most one, its powers satisfyA positive frequency-uniform gap therefore gives a uniform bound on every power. The Parseval identity then supplies spatial L2 norm stability. When the gap closes at a repeated unit root, a Jordan block can produce growth proportional to the number of steps despite both roots having modulus one.