Simultaneous permutations · standard sector — certificate

In the polynomial ring $R=\mathbb C[X_0,X_1,X_2,Y_0,Y_1,Y_2]$ in six algebraically independent variables, corresponding in order to indices $0,1,2,3,4,5$, fix the following polynomial:

Set $A=X_0^3Y_0^3-X_1^3Y_1^3$. With the fixed natural-number parameters $n=5$, $d=6$, and $q=1$, the theorem asserts that there exist polynomials $D,Q,N_0,N_1,N_2,N_3,N_4,N_5\in R$ such that all of the following hold simultaneously: $F\ne0$; every monomial with nonzero coefficient in $F$ has total degree $6$; every monomial with nonzero coefficient in $A$ has total degree $6$; $D\ne0$; writing $e=\deg D$ for the total degree of $D$, every monomial with nonzero coefficient in $D$ has total degree $e$; for each index $i\in\{0,1,2,3,4,5\}$, every monomial with nonzero coefficient in $N_i$ has total degree $e+1$; every polynomial dividing both $F$ and $D$ is a nonzero constant; and the following exact identity holds in $R$:

Here $(z_0,z_1,z_2,z_3,z_4,z_5)=(X_0,X_1,X_2,Y_0,Y_1,Y_2)$, and the partial derivatives are formal polynomial derivatives. The homogeneity conditions concern total monomial degree and allow any $N_i$ to be zero. The nonnegative integer $e$ is unrestricted, so a nonzero constant denominator is allowed. There is no separately imposed homogeneity or degree condition on $Q$, and $Q=0$ is allowed. All coefficients of the existentially quantified polynomials may be complex. The assertion is existence, with no uniqueness claim and no additional hypotheses.