  A classical result of A. Fleck states that if p is a prime, and n>0 and r are
integers, then $$\sum_{k=r(mod p)}\binom {n}{k}(-1)^k=0 (mod
p^{[(n-1)/(p-1)]}).$$ Recently R. M. Wilson used Fleck's congruence and
Weisman's extension to present a useful lemma on polynomials modulo prime
powers, and applied this lemma to reprove the Ax-Katz theorem on solutions of
congruences modulo p and deduce various results on codewords in p-ary linear
codes with weights. In light of the recent generalizations of Fleck's
congruence given by D. Wan, and D. M. Davis and Z. W. Sun, we obtain new
extensions of Wilson's lemma and the Ax-Katz theorem.
