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Conjectures on the distribution of roots modulo a prime of a polynomial

Yoshiyuki Kitaoka

Abstract

For a given monic integral polynomial $f(x)$ of degree $n$, we define local roots $r_i$ of $f(x)$ for a completely decomposable prime $p$ by $r_i \in \mathbb{Z}$, $f(r_i) \equiv 0 \bmod p$ and $0 \le r_1 \le r_2 \le \dots \le r_n < p$. With numerical data, we propose a conjecture on the distribution of $(r_1/p,\dots,r_n/p)$, which is a new kind of equi-distribution, and a conjecture of the distribution of $(r_1,\dots,r_n)$ which satisfies $r_i \equiv R_i \bmod L$ for given natural numbers $L,R_1,\dots,R_n$, which is nothing but Dirichlet's theorem on an arithmetic progression in the case $n = 1$.

Conjectures on the distribution of roots modulo a prime of a polynomial

Abstract

For a given monic integral polynomial of degree , we define local roots of for a completely decomposable prime by , and . With numerical data, we propose a conjecture on the distribution of , which is a new kind of equi-distribution, and a conjecture of the distribution of which satisfies for given natural numbers , which is nothing but Dirichlet's theorem on an arithmetic progression in the case .

Paper Structure

This paper contains 27 sections, 69 theorems, 557 equations.

Table of Contents

  1. Introduction and Conjectures
  2. Decimal expansion of a rational number
  3. Linear relation among roots
  4. Non-trivial linear relation
  5. Polynomial of $\deg=4$
  6. Abelian polynomial
  7. $S_3$
  8. Dependency of $Pr(f,\sigma),\frak{D}(f,\sigma)$ on $\sigma$
  9. $Pr(f,\sigma)$
  10. $\mathfrak{D}(f,\sigma)$
  11. One-dimensional distribution
  12. Case without non-trivial linear relation among roots
  13. Case of degree $4$ with non-trivial linear relation among roots
  14. More general case
  15. The Weyl criterion
  16. ...and 12 more sections

Key Result

Theorem 2.1

Let $a,b$ be natural numbers satisfying $(10a,b)=1$ and $a<b$, and let the purely periodic decimal expansion of $a/b$ be where $e$ is the minimal length of periods, i.e. $e =$ the order of $10 \bmod b$. Suppose $e = ln$ for natural numbers $n\,(>1),l$, and define natural numbers $B,L$ by $L=(10^l-1,b),\,b=BL$. We divide the period to $n$ parts of equal length $l$ , and add them; then we have whe

Theorems & Definitions (75)

  • Conjecture 1
  • Conjecture 2
  • Conjecture 3
  • Conjecture 4
  • Conjecture 5
  • Theorem 2.1
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • ...and 65 more