A statistical relation of roots of a polynomial in different local fields

Kitaoka, Yoshiyuki

doi:10.1090/s0025-5718-08-02133-9

Cited by 2 publications

(9 citation statements)

References 1 publication

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“…. , n) and suppose a linear relation (3). We may suppose that m = 0 in (3) to discuss the non-triviality of a linear relation, if necessary.…”

Section: Linear Relation Among Rootsmentioning

confidence: 99%

“…. , α p be roots of f, and suppose a linear relation (3). Adding a trivial relation α i = tr (f ) to (3) if necessary, we may assume that m i = 0.…”

Section: T)mentioning

confidence: 99%

“…In [3], there are examples of a polynomial of type 2, 3, but at that time the author did not recognize any S 3 -type polynomial satisfying the last condition in Proposition 5. In the next section, we give examples.…”

Section: ä ññ 2º If There Is a Non-trivial Relationmentioning

confidence: 99%

“…Because, they must be satisfied if d is the common denominator of Pr (f, L, {R i }) and an approximation by Pr X (f, L, {R i }) is sufficiently well. We consider the following four measures, abbreviating Pr X (f, [3]L, {R i }) to Pr X,R i : They are irreducible and indecomposable and define the same cyclotomic field Q(ζ 7 ), and the type of f 1 , f 2 is 2 and that of…”

Section: Expectation 2 For a Polynomial Of Degreementioning

confidence: 99%

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

Kitaoka¹

2017

Uniform Distribution Theory

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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 .

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“…. , n) and suppose a linear relation (3). We may suppose that m = 0 in (3) to discuss the non-triviality of a linear relation, if necessary.…”

Section: Linear Relation Among Rootsmentioning

confidence: 99%

“…. , α p be roots of f, and suppose a linear relation (3). Adding a trivial relation α i = tr (f ) to (3) if necessary, we may assume that m i = 0.…”

Section: T)mentioning

confidence: 99%

Section: ä ññ 2º If There Is a Non-trivial Relationmentioning

confidence: 99%

Section: Expectation 2 For a Polynomial Of Degreementioning

confidence: 99%

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

Kitaoka¹

2017

Uniform Distribution Theory

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“…For p ∈ Spl(f ), the definition of roots r i with (1) clearly implies that a n−1 + r 1 + · · · + r n = C p (f )p (2) for an integer C p (f ). The author has previously studied the statistical distribution of C p (f ) and local roots r i for p ∈ Spl(f ) ( [4]- [6], [8], [9]). A basic fact that we need here is as follows.…”

Section: Introductionmentioning

confidence: 99%

Statistical Distribution of Roots of a Polynomial Modulo Primes III

Kitaoka¹

2017

IJSP

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Let f (x) = x n + a n−1 x n−1 + · · · + a 0 (a n−1 , . . . , a 0 ∈ Z) be a polynomial with complex roots α 1 , . . . , α n and suppose that a linear relation over Q among 1, α 1 , . . . , α n is a multiple of ∑ i α i + a n−1 = 0 only. For a prime number p such that f (x) mod p has n distinct integer roots 0 < r 1 < · · · < r n < p, we proposed in a previous paper a conjecture that the sequence of points (r 1 /p, . . . , r n /p) is equi-distributed in some sense. In this paper, we show that it implies the equi-distribution of the sequence of r 1 /p, . . . , r n /p in the ordinary sense and give the expected density of primes satisfying r i /p < a for a fixed suffix i and 0 < a < 1.

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A statistical relation of roots of a polynomial in different local fields

Abstract: Abstract. Let f (x) be a monic polynomial in Z [x]. We observe a statistical relation of roots of f (x) in different local fields Q p , where f (x) decomposes completely. Based on this, we propose several conjectures.

Cited by 2 publications

References 1 publication

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

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

Statistical Distribution of Roots of a Polynomial Modulo Primes III

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