Positive characteristic approach to weak kernel conjecture

Abstract

When the polynomials f1; . . . ; fn A C1⁄2x1; . . . ; xn satisfy the Jacobian condition det qfi qxj i; j A C , the Kernel Conjecture says that Ker q qfn should be C1⁄2 f1; . . . ; fn 1 . In this paper, we prove a weaker version: When the leading monomials LMð f1Þ; . . . ;LMð ftÞ of f1; . . . ; ft (under a given monomial ordering) are linearly independent, then 7 i>t Ker q qfi 1⁄4 C1⁄2 f1; . . . ; ft . The main tool is the higher derivations q 1⁄2L fi , which behave like 1 L! q qfi L , but are defined for any rings, including positive characteristic ones. We reduce the problem of calculating the (higher) derivation kernels to the positive characteristic case, where we have a better control.