Polynomial Extension of Fleck’s Congruence

Extensions of Wilson ’ S Lemma and the Ax

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 ...

Extensions of Wilson ’ S Lemma and the Ax - Katz Theorem

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 ...

Fleck’s binomial congruence using circulant matrices

Recent Progress on Congruences Involving Binomial Coefficients

In 1913 A. Fleck proved that if p is a prime, and n > 0 and r are integers then ∑ k≡r (mod p) (n k ) (−1) ≡ 0 ( mod pb(n−1)/(p−1)c ) . Only recently the significance of Fleck’s congruence was realized. It plays a fundamental role in Colmez’ and Wan’s investigation of the ψ-operator related to Fontaine’s theory and p-adic Langlands correspondence. In this talk we give a survey of the recent deve...

Polynomial Extension of Fleck ’ S Congruence 3

Let p be a prime, and let f (x) be an integer-valued polynomial. By a combinatorial approach, we obtain a nontrivial lower bound of the p-adic order of the sum k≡r (mod p β) n k (−1) k f k − r p α , where α β 0, n p α−1 and r ∈ Z. This polynomial extension of Fleck's congruence has various backgrounds and several consequences such as k≡r (mod p α) n k a k ≡ 0 mod p n−p α−1 ϕ(p α) provided that ...