Four plus two · suspension remainder — certificate

The theorem has no free parameters or hypotheses and asserts the following existence statement. In $R=\mathbb C[z_0,z_1,z_2,z_3,z_4,z_5]$, with variable $z_i$ corresponding to index $i\in\{0,1,2,3,4,5\}$, fix $n=5$, $d=6$, $q=1$, $A=z_0^3z_3^3$, and

There exist polynomials $D,Q\in R$ and an indexed family $N_0,\ldots,N_5\in R$ such that $F\ne0$; every monomial with nonzero coefficient in $F$ has total degree $6$; every monomial with nonzero coefficient in $A$ has total degree $6(1+1)-(5+1)=6$; $D\ne0$; writing $t=\deg D$ for its total degree, every monomial with nonzero coefficient in $D$ has total degree $t$; for each $i=0,\ldots,5$, every monomial with nonzero coefficient in $N_i$ has total degree $t+6\cdot1-5=t+1$; every polynomial dividing both $F$ and $D$ is a unit of $R$ (equivalently, every common divisor is a nonzero complex constant); and the following equality holds as a polynomial identity in $R$:

Here a monomial $z_0^{e_0}\cdots z_5^{e_5}$ has total degree $e_0+\cdots+e_5$, with all $e_i$ nonnegative integers. The degree expressions for $A$ and $N_i$ are evaluated in the integers. The nonzero homogeneous denominator $D$ may have any nonnegative degree, including degree zero; no upper degree bound is imposed. Each $N_i$ may be zero, since its coefficient condition is then vacuous. No degree, homogeneity, or nonzero condition is imposed on $Q$, and $Q=0$ is allowed but not required. The assertion includes the stated nonzero and homogeneity properties of the fixed polynomials as conjuncts of the existential conclusion.