What does this research mean for the field?

The distribution of polynomials evaluated at the roots of a fully decomposable monic integral polynomial modulo a prime follows specific conjectural patterns. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To investigate the geometric and statistical distribution of ordered roots modulo a prime for monic integral polynomials that fully decompose modulo that prime.

September 5, 2026MathematicsOpen Access

Notes on the Distribution of Roots Modulo a Prime of a Polynomial VI: Case of Polynomials with Local Roots

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To investigate the geometric and statistical distribution of ordered roots modulo a prime for monic integral polynomials that fully decompose modulo that prime.
Analyzed monic integral polynomials of degree n that split completely into linear factors modulo a prime p.
Examined the spatial arrangement of ordered root tuples (r1, ..., rn) satisfying 0 ≤ r1 ≤ ··· ≤ rn < p.
Formulated several theoretical conjectures detailing the distribution of polynomial expressions evaluated at the roots modulo p.
Identified distribution characteristics specific to classes of polynomials possessing local roots.

Implication

Theoretical analysis reveals distribution patterns of roots modulo a prime, highlighting new conjectures for polynomials with local roots.

Abstract

Let f(x) be a monic integral polynomial of degree n and p a prime number for which f(x) is fully decomposable modulo p. Let integers r1,...,rn be the roots of f(x) mod p with 0 ≤ r1 ≤ ··· ≤ rn < p. Inthis series of papers, we have investigated the distribution of the points (r1,. . .,rn). In the present paper, we present several conjectures concerning the distribution of polynomials in the roots ri.

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Cite This Study

Yoshiyuki Kitaoka (2026) studied this question.

synapsesocial.com/papers/6a9bd3a66b95aff0620eac77 https://doi.org/10.3390/math14173175

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Implication

Theoretical analysis reveals distribution patterns of roots modulo a prime, highlighting new conjectures for polynomials with local roots.

Notes on the Distribution of Roots Modulo a Prime of a Polynomial VI: Case of Polynomials with Local Roots

Key Points

Implication

Abstract

Discussion

Cite This Study

Notes on the Distribution of Roots Modulo a Prime of a Polynomial VI: Case of Polynomials with Local Roots

Key Points

Implication

Abstract

Discussion

Cite This Study

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