Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity

Let f(x) be a non-zero polynomial with integer coefficients. An automor phism ? of a group G is said to satisfy the elementary abelian identity f(x) if the linear transformation induced by ? on every characteristic elementary abelian section of G is annihilated by f(x). We prove that if a finite (soluble) group G admits a fixed-point-free automorphism ? satisfying an elementary abelian identity f(x), where f(x) is a primitive polynomial, then the Fitting height of G is bounded in terms of deg(f(x)). We also prove that if f(x) is any non-zero polynomial and G is a ? 0 -group for a finite set of primes ? = ?(f(x)) depending only on f(x), then the Fitting height of G is bounded in terms of the number irr(f(x)) of different irreducible factors in the decomposition of f(x). These bounds for the Fitting height are stronger than the well-known bounds in terms of the composition length ?(|?|) of h?i when deg f(x) or irr(f(x)) is small in comparison with ?(|?|).