Cyclic coupling · pole order two — certificate

In the polynomial ring $R=\mathbb C[z_0,z_1,z_2,z_3,z_4,z_5]$, fix $F=z_0^6+z_1^6+z_2^6+z_3^6+z_4^6+z_5^6+\tfrac12 z_0^3z_1z_3z_4+\tfrac12 z_0z_2^3z_3z_5+\tfrac12 z_1^3z_2z_4z_5$ and $A=z_0^2z_1^2z_2^3z_3^2z_4^2z_5$. With no hypotheses or variable parameters, the theorem asserts that there exist polynomials $D,Q\in R$ and a family of polynomials $(N_i)_{i=0}^{5}$ in $R$ satisfying all of the following conditions: $F\ne0$; every monomial of $F$ with nonzero coefficient has total degree $6$; every monomial of $A$ with nonzero coefficient has total degree $12$; $D\ne0$; writing $e=\deg D$ for the total degree of $D$, every monomial of $D$ with nonzero coefficient has total degree $e$; for each $i\in\{0,1,2,3,4,5\}$, every monomial of $N_i$ with nonzero coefficient has total degree $e+7$; every polynomial that divides both $F$ and $D$ is a unit of $R$, equivalently a nonzero constant; and the polynomial identity $AD^2-2D\sum_{i=0}^{5}(\partial F/\partial z_i)N_i+F\bigl(D\sum_{i=0}^{5}\partial N_i/\partial z_i-\sum_{i=0}^{5}(\partial D/\partial z_i)N_i\bigr)=F^3Q$ holds in $R$, where the derivatives are formal partial derivatives. Here $e$ is a nonnegative integer with no prescribed upper bound, and a nonzero constant denominator, for which $e=0$, is allowed. Each $N_i$ is permitted to be zero, in which case its condition on nonzero monomials is vacuous. The polynomial $Q$ may also be zero and has no separately imposed degree or homogeneity condition. The witnesses are asserted to exist, with no uniqueness requirement.