Congruences de sommes de chiffres de valeurs polynomiales

Let m, g, q ∈ N with q 2 and (m, q − 1) = 1. For n ∈ N, denote by sq(n) the sum of digits of n in the q-ary digital expansion. Given a polynomial f with integer coefficients, degree d 1, and such that f(N) ⊂ N, it is shown that there exists C = C(f, m, q) > 0 such that for any g ∈ Z, and all large N, |{0 n N : sq(f(n)) ≡ g (mod m)}| CN min(1,2/d!) . In the special case m = q = 2 and f(n )= n 2 , the value C =1 /20 is admissible. Classification AMS : principale 11B85, secondaires 11N37, 11N69.