Positive characteristic approach to Weak Kernel Conjecture

概要

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When the polynomials $f_1, \ldots, f_n\in \cc[x_1, \ldots, x_n]$ satisfy the Jacobian condition $\det \left( \left( \frac{\partial f_i}{\partial x_j}\right)_{i, j}\right)\in \cc^*$, the Kernel Conjecture says that $\Ker \frac{\partial}{\partial f_n}$ should be $\cc[f_1, \ldots, f_{n-1}]$. In this paper, we prove a weaker version: When the leading monomials ${\rm LM}(f_1), \ldots, {\rm LM} (f_t)$ of $f_1, \ldots f_t$ (under a given monomial ordering) are linearly independent, then $\displaystyle \cap_{i>t}\Ker \left(\frac{\partial}{\partial f_i}\right)=\cc[f_1, \ldots, f_t]$. The main tool is the higher derivations $\partial_{f_i}^{[L]}$, which behave like $\frac{1}{L!}\left(\frac{\partial}{\partial f_i}\right)^L$, 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.
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