论文标题
在单位值多项式函数上
On the group of unit-valued polynomial functions
论文作者
论文摘要
令$ r $为有限的交换戒指，$ 1 \ ne 0 $。 $ r $上多项式函数的集合$ \ MATHCAL {F}（R）$是带有尖端操作的有限交换环。它的组$ \ Mathcal {f}（r）^\ times $只是所有单位值多项式函数的集合，这是将$ r $映射到其组单元组的一组多项式函数。我们表明，$ \ Mathcal {p} _r（r [x]/（x^2））$ ring $ r [x]/（x^2）$上的多项式排列组，由$ r $以上的$ r $代表的排列组成，在$ \ nathcal $ \ nathcal的半级产品中（ $ \ Mathcal {p}（r）$ $ r $上的多项式排列组。特别是，当$ r = \ mathbb {f} _q $时，我们证明了$ \ mathcal {p} _ {\ Mathbb {f} _q}（\ Mathbb {f} _q [x] _q [x]/（x]/（x^2）） \ltimes_θ\ Mathcal {f}（\ Mathbb {f} _Q）^\ times $。此外，我们计算单位值多项式函数$ \ pmod {p^n} $，并获得这些功能的规范表示。
Let $R$ be a finite commutative ring with $1\ne 0$. The set $\mathcal{F}(R)$ of polynomial functions on $R$ is a finite commutative ring with pointwise operations. Its group of units $\mathcal{F}(R)^\times$ is just the set of all unit-valued polynomial functions, that is the set of polynomial functions which map $R$ into its group of units. We show that $\mathcal{P}_R(R[x]/(x^2))$ the group of polynomial permutations on the ring $R[x]/(x^2)$, consisting of permutations represented by polynomials over $R$, is embedded in a semidirect product of $\mathcal{F}(R)^\times$ by $\mathcal{P}(R)$ the group of polynomial permutations on $R$. In particular, when $R=\mathbb{F}_q$, we prove that $\mathcal{P}_{\mathbb{F}_q}(\mathbb{F}_q[x]/(x^2))\cong \mathcal{P}(\mathbb{F}_q) \ltimes_θ\mathcal{F}(\mathbb{F}_q)^\times$. Furthermore, we count unit-valued polynomial functions $\pmod{p^n}$ and obtain canonical representations for these functions.
