Scattered polynomials over a finite field ${\mathbb{F}}_{q^n}$ have been introduced by Sheekey in 2016, and a central open problem regards the classification of those that are exceptional. So far, only two families of exceptional scattered polynomials are known. Very recently, Longobardi and Zanella weakened the property of being scattered by introducing the notion of L-$q^t$-partially scattered and R-$q^t$-partially scattered polynomials, for t a divisor of n. Indeed, a polynomial is scattered if and only if it is both L-$q^t$-partially scattered and R-$q^t$-partially scattered. In this paper, by using techniques from algebraic geometry over finite fields and function fields theory, we show that the property which is the hardest to be preserved is the L-$q^t$-partially scattered one. We investigate a large family $\mathcal{F}$ of R-$q^t$-partially scattered polynomials, containing examples of exceptional R-$q^t$-partially scattered polynomials, which turn out to be connected with linear sets of so-called pseudoregulus type. We introduce two different notions of equivalence preserving the property of being R-$q^t$-partially scattered. Many polynomials in $\mathcal{F}$ are inequivalent and geometric arguments are used to determine their equivalence classes under the action of $\mathrm{\Gamma }\mathrm{L}(2n/t,q^t)$.

有限域上的分散多项式 ${\mathbb{F}}_{q^n}$Sheekey 于 2016 年引入，一个核心的开放问题是关于那些异常的分类。到目前为止，只有两个异常分散多项式家族是已知的。最近，Longobardi 和 Zanella 通过引入L-$q^{吨}$-部分分散和R-$q^{吨}$- 部分分散多项式，因为t是n的除数。事实上，一个多项式是分散的当且仅当它同时是 L-$q^{吨}$-部分分散和R-$q^{吨}$- 部分分散。在本文中，通过使用有限域上的代数几何技术和函数域理论，我们证明了最难保持的性质是 L-$q^{吨}$- 部分分散的一个。我们调查一个大家庭$\mathcal{F}$ R-$q^{吨}$- 部分分散多项式，包含异常 R 的例子-$q^{吨}$- 部分分散的多项式，结果证明它们与所谓的伪正则类型的线性集有关。我们引入了两种不同的等价概念，保留了 R-$q^{吨}$- 部分分散。许多多项式在$\mathcal{F}$ 是不等价的，几何参数用于确定它们在作用下的等价类 $\mathrm{\Gamma }\mathrm{升}(2n/吨,q^{吨})$.