Translation invariants of Fourier-mode phases ($n\cdot \theta ,{\sum }_j{n}_j{q}_j=0$)

If a translation symmetry acts on complex Fourier modes by $A_j\mapsto e^{i{q}_j\cdot s}{A}_j$, a linear combination of their phases is invariant precisely when its coefficient vector lies in the kernel of the matrix of wavevectors. For four amplitudes whose wavevectors span a plane, this leaves two independent phase variables after quotienting translations. At zero amplitude, polar phases are undefined, so smooth centre manifold equations should first be written in Cartesian amplitudes.

An equivariant dynamical system can contain the monomial ${\prod }_j{A}_j^{p_j}{\overline{A}}_j^{q_j}$ in its $\ell$th amplitude equation only if its total wavevector equals the output wavevector. Its degree is ${\sum }_j(p_j+q_j)$. Thus an integer vector $d=p-q$ contributes at degree $N$ exactly when the wavevector equation holds, $\Vert d{\Vert }_1\le N$, and $N-\Vert d{\Vert }_1$ is even. The extra even degree is supplied by factors $|A_j{|}^2$. This finite integer test distinguishes regular saturation terms from phase-sensitive resonance terms.