Broken paired signs — certificate

The declaration asserts the following existence statement in the polynomial ring $R=\mathbb C[z_0,z_1,z_2,z_3,z_4,z_5]$, with no universally quantified parameters or hypotheses. Fix $F=\sum_{i=0}^{5}z_i^6+\tfrac12 z_0^3z_1z_3z_4+\tfrac13 z_0^2z_1z_2z_3z_5+\tfrac12 z_1^3z_2z_4z_5$ and $A=z_0^3z_3^3$; the numerical parameters are $n=5$, $d=6$, and $q=1$. There exist polynomials $D,Q\in R$ and a family $(N_i)_{i=0}^{5}$ of polynomials in $R$ such that all of the following hold simultaneously: $F\ne0$; $F$ is homogeneous of degree $6$; every monomial of $A$ with nonzero coefficient has total degree $6$; $D\ne0$; $D$ is homogeneous of degree $r$, where $r=\operatorname{totalDegree}(D)$; for every $i\in\{0,1,2,3,4,5\}$, every monomial of $N_i$ with nonzero coefficient has total degree $r+1$; every polynomial dividing both $F$ and $D$ is a unit of $R$, that is, a nonzero constant polynomial; and the polynomial identity $A D^2-D\sum_{i=0}^{5}\frac{\partial F}{\partial z_i}N_i+F\left(D\sum_{i=0}^{5}\frac{\partial N_i}{\partial z_i}-\sum_{i=0}^{5}\frac{\partial D}{\partial z_i}N_i\right)=F^2Q$ holds in $R$. Here all partial derivatives are formal polynomial derivatives, and the total degree of a monomial $z_0^{a_0}\cdots z_5^{a_5}$ is $a_0+\cdots+a_5$, with each $a_i$ a nonnegative integer. The degree conditions on $A$ and $N_i$ quantify over every such exponent tuple whose coefficient is nonzero; they impose no requirement at tuples with zero coefficient, and they permit any $N_i$ to be the zero polynomial. The integer degree expressions specialize to $6(1+1)-(5+1)=6$ for $A$ and $r+6\cdot1-5=r+1$ for each $N_i$, so neither specified degree is negative. The degree $r$ has no prescribed upper bound and may be $0$, so a nonzero constant denominator is included among the possible witnesses. The witness $Q$ is an arbitrary polynomial subject to the displayed identity, with no separately imposed degree, homogeneity, or nonzero condition; in particular, $Q=0$ is permitted. The witnesses are asserted to exist, with no uniqueness requirement.