Let u1, . . . , um be linear recurrences with values in a field K of positive characteristic p. We show that the set of integer vectors (k1, . . . , km) such that u1(k1) + · · ·+ um(km) = 0 is p-normal in a natural sense generalizing that of the first author, who proved the result for m = 1. Furthermore the set is effectively computable if K is. We illustrate this with an example for m = 4. We ...