Congruences, Trees, and 𝑝-adic Integers

Abstract

Let f f be a polynomial in one variable with integer coefficients, and p p a prime. A solution of the congruence f ( x ) ≡ 0 ( mod p ) f(x) \equiv 0 (\text {mod} \,p) may branch out into several solutions modulo p 2 p^{2} , or it may be extended to just one solution, or it may not extend to any solution. Again, a solution modulo p 2 p^{2} may or may not be extendable to solutions modulo p 3 p^{3} , etc. In this way one obtains the “solution tree” T = T ( f ) T = T(f) of congruences modulo p λ p^{\lambda } for λ = 1 , 2 , … \lambda = 1,2,\ldots . We will deal with the following questions: What is the structure of such solution trees? How many “isomorphism classes” are there of trees T ( f ) T(f) when f f ranges through polynomials of bounded degree and height? We will also give bounds for the number of solutions of congruences f ( x ) ≡ 0 ( mod p λ ) f(x) \equiv 0 (\text {mod} \,p^{\lambda }) in terms of p , λ p, \lambda and the degree of f f .