We prove that the local $\mathbb{A}^1$-degree of a polynomial function at an isolated zero with finite separable residue field is given by the trace of the local $\mathbb{A}^1$-degree over the residue field. This fact was originally suggested by Morel's work on motivic transfers and by Kass and Wickelgren's work on the Scheja-Storch bilinear form. As a corollary, we generalize a result of Kass and Wickelgren's relating the Scheja-Storch form and the local $\mathbb{A}^1$-degree.

中文翻译：

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局部$\mathbb{A}^1$-degree的踪迹

我们证明了局部 $\mathbb{A}^1$-degree 多项式函数在具有有限可分残差域的孤立零处的局部 $\mathbb{A}^1$-degree 由局部 $\mathbb{A}^1$-degree 在残场。这个事实最初是由 Morel 关于动机转移的工作以及 Kass 和 Wickelgren 关于 Scheja-Storch 双线性形式的工作提出的。作为推论，我们概括了 Kass 和 Wickelgren 将 Scheja-Storch 形式与局部 $\mathbb{A}^1$-degree 相关联的结果。