In this paper, two types of permutation polynomials, ${(x^{2^m}+x+\delta )}^s+cx$ and $x^r(x^{q-1}+a)$, are studied. To formulate the necessary and sufficient conditions for the polynomials to be permutation polynomials over finite fields, the structures and properties of the field elements are analyzed. Meanwhile, the number of solutions to some equations over finite fields is investigated. For quadratic equations, the necessary and sufficient conditions are utilized thoroughly to study the permutation polynomials. But for cubic equations, only one direction is investigated by the existing literatures. This paper employs the properties of these equations in full strength to study ${(x^{2^m}+x+\delta )}^s+cx$ over ${\mathbb{F}}_{2^{3m}}^{⁎}$. For the binomials of the form $x^r(x^{q-1}+a)$, two different methods are exploited to formulate the necessary and sufficient conditions over ${\mathbb{F}}_{q^3}$, and some partial results are obtained for ${\mathbb{F}}_{q^e}$.

在本文中，两种类型的置换多项式， ${(X^{2^{米}}+X+\delta )}^{秒}+CX$ 和 $X^r(X^{q-1}+\mathrm{一种})$，被研究。为了制定多项式为有限域上置换多项式的充要条件，分析了场元的结构和性质。同时，研究了一些方程在有限域上的解数。对于二次方程，充分利用充分必要条件来研究置换多项式。但对于三次方程，现有文献只研究了一个方向。本文充分利用这些方程的性质来研究${(X^{2^{米}}+X+\delta )}^{秒}+CX$ 超过 ${\mathbb{F}}_{2^{3米}}^{⁎}$. 对于形式的二项式$X^r(X^{q-1}+\mathrm{一种})$, 两种不同的方法被用来制定必要和充分条件 ${\mathbb{F}}_{q^3}$，并获得了一些部分结果 ${\mathbb{F}}_{q^{\mathrm{电子}}}$.