Dense rational · full coefficient support — certificate

The declaration has no variable parameters or hypotheses. Work in the polynomial ring $R=\mathbb C[z_0,z_1,z_2,z_3,z_4,z_5]$, with the fixed natural numbers $n=5$, $d=6$, $q=1$, and $A=z_0^3z_3^3$. To specify the fixed polynomial $F$ completely, put $x=(z_0,z_1,z_2)$ and $y=(z_3,z_4,z_5)$, and, for $m\in\{2,4,6\}$ and an ordered triple $t=(t_0,t_1,t_2)$, let $M_m(t)$ be the row vector of all monomials $t_0^at_1^bt_2^c$ with $a,b,c\in\mathbb N$ and $a+b+c=m$, ordered first by decreasing $a$ and then, for each $a$, by decreasing $b$; thus it has $(m+1)(m+2)/2$ entries. The polynomial is

where the vectors and matrices, in precisely those row and column orders, are

The assertion is that there exist polynomials $D,Q\in R$ and a family $(N_i)_{i=0}^{5}$ of polynomials in $R$ satisfying all the following conditions simultaneously: $F\ne0$; every monomial of $F$ with nonzero coefficient has total degree $6$; every monomial of $A$ with nonzero coefficient has total degree $6$; $D\ne0$; if $e$ denotes the total degree of $D$, every monomial of $D$ with nonzero coefficient has total degree $e$; for every $i\in\{0,1,2,3,4,5\}$, every monomial of $N_i$ with nonzero coefficient has total degree $e+1$; every common polynomial divisor of $F$ and $D$ is a unit of $R$, that is, a nonzero complex constant; and the following is an equality in $R$, where $\partial_i$ denotes the formal partial derivative with respect to $z_i$:

The witnesses may have arbitrary complex coefficients. The degree $e$ has no upper bound and may be $0$, allowing a nonzero constant $D$. Each zero $N_i$ satisfies its monomial-degree condition vacuously, and $Q$ may be zero; these possibilities remain subject to the displayed identity and the other conditions. There is no separate homogeneity, degree, or nonvanishing requirement on $Q$, and the assertion requires existence, not uniqueness, of the witnesses.