DISCRIMINANT VALUES OF INTEGRAL POLYNOMIALS IN THE ARCHIMEDEAN AND NON-ARCHIMEDEAN METRICS

Abstract

Consider the class 3(Q) of the polynomials P(t)∈[t] of degree 3 and height H(P) ≤ Q, Q >1. Define a subclass S3(Q) in 3(Q) by taking the polynomials P(t) having discriminants not exceeding Q2n−2−2v1 and divisible by the power of the prime number pe , pe > Q2v2 , v1 ≥ 0, v2 ≥ 0, 0 ≤ v1 + v2 < 3 / 2. The upper bound on the number of the elements in S3(Q). is found. It has been proved that for any ε > 0 and Q > Q0 (ε), the inequality 4 5/3( 1 2 ) # 3( ) S Q Q v v < − + +ε is valid.