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Scattered polynomials over a finite field $\mathbb{F}_{q^n}$ have beenintroduced by Sheekey in 2016, and a central open problem regards theclassification of those that are exceptional. So far, only two families ofexceptional scattered polynomials are known. Very recently, Longobardi andZanella weakened the property of being scattered by introducing the notion ofL-$q^t$-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 isboth L-$q^t$-partially scattered and R-$q^t$-partially scattered. In thispaper, by using techniques from algebraic geometry over finite fields andfunction fields theory, we show that the property which is is the hardest to bepreserved is the L-$q^t$-partially scattered one. On the one hand, we are ableto extend the classification results of exceptional scattered polynomials toexceptional L-$q^t$-partially scattered polynomials. On the other hand, theR-$q^t$-partially scattered property seems more stable. We present a largefamily of R-$q^t$-partially scattered polynomials, containing examples ofexceptional R-$q^t$-partially scattered polynomials, which turn out to beconnected with linear sets of so-called pseudoregulus type. In order to detectnew examples of polynomials which are R-$q^t$-partially scattered, we introducetwo different notions of equivalence preserving this property and concerningnatural actions of the groups ${\rm \Gamma L}(2,q^n)$ and ${\rm \GammaL}(2n/t,q^t)$. In particular, our family contains many examples of inequivalentpolynomials, and geometric arguments are used to determine the equivalenceclasses under the action of ${\rm \Gamma L}(2n/t,q^t)$.

Investigating the exceptionality of scattered polynomials