一类一般次数多项式代数的平方运算与命中问题
The squaring operation and the hit problem for the polynomial algebra in a type of generic degree
摘要：
设$P_k$是分次多项式代数$\mathbb F_2[x_1,x_2,\ldots ,x_k]$，每个生成元$x_i$的次数为1，其中$\mathbb F_2$表示含有两个元素的素域. Frank Peterson的命中问题要求多项式代数$P_k$作为模在mod-2 Steenrod代数$\mathcal{A}$上的一个极小生成集.等价地，我们想在每个度上为$\mathbb F_2 \otimes_{\mathcal A} P_k$找到一个向量空间基。在本文中，我们研究了Kameko平方运算$\widetilde{Sq}^0_*: \mathbb F_2 \otimes_{\mathcal A} P_k \longrightarrow \mathbb F_2 \otimes_{\mathcal A} P_k$的核的生成集在所谓的一般度。通过使用这个结果，我们明确地计算击中问题的$k=5$在各自的通用程度。
英文摘要：
Let $P_k$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ with the degree of each generator $x_i$ being 1, where $\mathbb F_2$ denote the prime field with two elements. The hit problem of Frank Peterson asks for a minimal generating set for the polynomial algebra $P_k$ as a module over the mod-2 Steenrod algebra $\mathcal{A}$. Equivalently, we want to find a vector space basis for $\mathbb F_2 \otimes_{\mathcal A} P_k$ in each degree. In this paper, we study a generating set for the kernel of Kameko's squaring operation $\widetilde{Sq}^0_*: \mathbb F_2 \otimes_{\mathcal A} P_k \longrightarrow \mathbb F_2 \otimes_{\mathcal A} P_k$ in a so-called generic degree. By using this result, we explicitly compute the hit problem for $k=5$ in the respective generic degree.
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