二项式的刚性条件

中文摘要 AI 辅助

我们考虑多元多项式中的以下三个条件。条件(1)对多项式成立，如果对于每个仿射整环，以下成立：每当将代数独立元素代入多项式的结果位于一个局部幂零导子的核中时，这些元素本身必须位于该核中。多项式满足条件(2)，如果它不属于多项式代数的任何非零LND的核。它满足条件(3)，如果商代数模该多项式生成的理想是刚性的，即它不允许非平凡的LND。对于不可约二项式，我们证明所有三个条件等价。对于可约二项式，我们建立了这些条件之间的所有现有蕴含关系。

英文摘要

We consider the following three conditions for polynomials in several variables. Condition \textbf{(1)} holds for a polynomial if, for every affine integral domain, the following is true: whenever the result of substituting algebraically independent elements into the polynomial lies in the kernel of a locally nilpotent derivation, the elements themselves must lie in that kernel. A polynomial satisfies condition \textbf{(2)} if it does not belong to the kernel of any nonzero LND of the polynomial algebra. It satisfies condition \textbf{(3)} if the quotient algebra modulo the ideal generated by the polynomial is rigid, that is, it admits no nontrivial LNDs. For irreducible binomials, we prove that all three conditions are equivalent. For reducible binomials, we establish all existing implications between these conditions.

发表机构

• HSE University（高等经济大学）

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