Injectivity of compressing maps on the set of primitive sequences modulo square-free odd integers

Let p(1), p(2),..., p(r) be distinct odd primes and m = p(1)p(2)(...)p(r). Let f (x) be a primitive polynomial of degree n over Z/mZ. Denote by L(f) the set of primitive linear recurring sequences generated by f (x). A map psi on Z/mZ naturally induces a map (psi) over cap on L(f), mapping a sequence (...,s(i -1), s(i), s(i +1),...) to (...,psi(s(i-1)),psi(s(i)),psi(s(i +1)),...). Previous results gave sufficient conditions under which modular functions induce injective maps on L(f). In this article we give an inequality which holds for large enough n. If this inequality holds, then the injectivity of (psi) over cap is clearly determined for any map psi on Z/mZ. Particularly, the modular function psi(a) = a mod M induces an injective map on L(f) for any M is an element of {2 <= i is an element of Z : i inverted iota m}.