The Generalized Vanishing Conjecture: The Two-Variable Theorem and the First Failing Dimension

We determine the exact dimensional range of the Generalized Vanishing Conjecture for constant-coefficient differential operators over fields of characteristic zero. We prove the conjecture in two variables for arbitrary, possibly nonhomogeneous, operators. The proof combines Hall's marriage theorem, Newton-face separation, and a prime-dilation argument; apart from the constant-term theorem of Duistermaat and van der Kallen, it is elementary and uses neither degree bounds nor computer algebra. We then construct a homogeneous counterexample in three variables. With rho=t^2+xy, A=rho+x^2, C=y rho^2-2x t^2 rho-x^3 t^2, Delta=4 partial_x partial_y+partial_t^2, Lambda=Delta^6, and P=AC^2, the paper proves that Lambda^m(P^m)=0 while Lambda^m(x^2 P^m) is nonzero for every m>=1. The construction extends to an explicit cusp-profile family and gives counterexamples for every Delta^k with k>=6. Consequently, the unrestricted Generalized Vanishing Conjecture, and already its homogeneous form, holds in dimension n exactly when n<=2.