论文标题
在有限交换的本地戒指上警告数字
Waring numbers over finite commutative local rings
论文作者
论文摘要
在本文中，我们研究$ g_r（k）$ for $（r，\ frak m）$ g_r（k）$ $ a \ kathbb {n} $带有$（k，| r |）= 1 $的有限交换本地戒指。我们首先将警告号$ g_r（k）$与cayley图的直径相关联$ g_r（k）= cay（r，u_r（k））$和$ w_r（k）= cay（r，s_r（k））$ with $ u_r（k）$ u_r（k） x \ in r^\ times \} $，区分图形指向或无方向性的情况。我们表明，在这两种情况下（指示或无向），可以通过炸毁$ g _ {\ Mathbb {f} _q} _q}（k）$ a数字$ | \ frak {m frak {m rak {m} | $的$ g _ { $ r/\ frak m $。然后，通过使用上述爆炸，我们将在本地环$ r $上缩小WARING NUMBER $ G_R（K）$的研究，以计算有限的剩余场上$ r/\ frak m \ simeq \ simeq \ simeq \ simeq \ mathbb {f} f} _q $ g（k，q）$。通过这种方式，使用有限字段的尾声的已知结果，我们获得了几个明确的结果，即具有有限交换的本地环，并获得了具有身份的有限的本地环。
In this paper we study Waring numbers $g_R(k)$ for $(R,\frak m)$ a finite commutative local ring with identity and $k \in \mathbb{N}$ with $(k,|R|)=1$. We first relate the Waring number $g_R(k)$ with the diameter of the Cayley graphs $G_R(k)=Cay(R,U_R(k))$ and $W_R(k)=Cay(R,S_R(k))$ with $U_R(k) = \{ x^k : x\in R^*\}$ and $S_R(k)=\{x^k : x\in R^\times\}$, distinguishing the cases where the graphs are directed or undirected. We show that in both cases (directed or undirected), the graph $G_R(k)$ can be obtained by blowing-up the vertices of $G_{\mathbb{F}_q}(k)$ a number $|\frak{m}|$ of times, with independence sets the cosets of $\frak{m}$, where $q$ is the size of the residue field $R/\frak m$. Then, by using the above blowing-up, we reduce the study of the Waring number $g_R(k)$ over the local ring $R$ to the computation of the Waring number $g(k,q)$ over the finite residue field $R/\frak m \simeq \mathbb{F}_q$. In this way, using known results for Waring numbers over finite fields, we obtain several explicit results for Waring numbers over finite commutative local rings with identity.
