Notes on the Distribution of Roots Modulo a Prime of a Polynomial

Abstract

ABSTRACT. Let f (x) be a monic polynomial in Z[x] with roots α 1 , . . . , α n . We point out the importance of linear relations among 1, α 1 , . . . , α n over rationals with respect to the distribution of local roots of f modulo a prime. We formulate it as a conjectural uniform distribution in some sense, which elucidates data in previous papers. Communicated by Shigeki AkiyamaIn this note, a polynomial means always a monic one over the ring Z of integers and the letter p denotes a prime number, unless specified. Letbe a polynomial of degree n. As in the previous papers, we putis fully splitting modulo p} for a positive number X and Spl (f ) := Spl ∞ (f ). In this note, we require the following conditions on the local roots r 1 , . . . , r n (∈ Z) of f (x) ≡ 0 mod p for a prime p ∈ Spl (f ) :We can determine local roots r i uniquely with this global ordering. If f is irreducible and of deg(f ) > 1, and p is sufficiently large, then (2) is equivalent to 0 < r 1 < · · · < r n < p. Here, we consider two types of distribution of local roots r i of f .

Notes on the Distribution of Roots Modulo a Prime of a Polynomial

“…is satisfied only if m 1 = • • • = m n . We know that if for an irreducible polynomial f the degree n is prime or the Galois group is isomorphic to the symmetric group S n , then there is only a trivial linear relation among roots [3]. We conjectured a kind of uniformity on the distribution of (r 1 /p, .…”

“…In the table, 1(3) for m = 7, n = 2 means 1. • • • × 10 −3 = 0.001 • • • and so on.Before a calculation of the volume of vol(D a ∩ Dn ), we refer to the following fundamental lemma ([1]).…”

Elementary Integration of Superelliptic Integrals

“…. Now, a single function LS is necessary (see Propositions 4,6). Trager's approach is to build a divisor on the superelliptic curve, and then to test its principality using linear algebra.…”

Uniform distribution of fractional parts of average of kth power of prime divisors

“…, which can be interpreted as the harmonic means of prime divisors of n taken with and without multiplicity, respectively. In 2009, Kátai and Luca [6] proved that the sequences (h(n)) and ( h(n)) are uniformly distributed. More generally, they proved that if f (n) is an additive function such that there exist two positive constants c 1 and c 2 such that f (p) < c 1 /p and 0 < f (p a ) < c 2 for all primes p and all positive integers a, then the sequences (ω(n)/f (n)) and (Ω(n)/f (n)) are uniformly distributed.…”