The nilpotency of finite groups with a fix-point-free automorphism satisfying an identity

We generalize the positive solution of the Frobenius conjecture and refinements thereof by studying the structure of groups that admit a fix-point-free automorphism satisfying an identity. We show, in particular, that for every polynomial $r(t) = a_0 + a_1 \cdot t + \cdots + a_d \cdot t^d \in $\mathbb{Z}$[t]$thatisirreducibleover$$\mathbb{Q}$$,thereexist(explicit)invariants$a,b,c \in $\mathbb{N}$$ with the following property. Consider a finite group with a fix-point-free automorphism $\alpha :G⟶G$ and suppose that for all $x\in G$ we have the equality $x^{a_0} \cdot \alpha(x^{a_1}) \cdot \alpha^2(x^{a_2})\cdots \alpha^d(x^{a_d}) = 1_G.$Then$G$issolvableandoftheform$A \cdot (B \rtimes (C \times D))$,where$A$isan$a$-group,$B$isa$b$-group,$C$ is a nilpotent $c$-group, and $D$ is a nilpotent group of class at most $d^{2^d}$. Here, a group $H$ is said to be an $a$-group (resp. $b$-group or $c$-group) if the order of every $h\in H$ divides some natural power of $a$ (resp. $b$ or $c$).