Linear equations over multiplicative groups, recurrences, and mixing II

Let u1,…,umu1,…,um be linear recurrences with values in a field KK of positive characteristic pp. We show that the set of integer vectors (k1,…,km)(k1,…,km) such that u1(k1)+⋯+um(km)=0u1(k1)+⋯+um(km)=0 is pp-normal in a natural sense generalizing that of the first author, who proved the result for m=1m=1. Furthermore the set is effectively computable if KK is. We illustrate this with an example for m=4m=4. We also show that the corresponding set for zero characteristic is not decidable for m=557844m=557844, thus verifying a conjecture of Cerlienco, Mignotte, and Piras.