Work

Integer Translates of a Monic Cubic: Resultants and Common Factors over Finite Fields

Year: 2026

Type: preprint

Abstract: We study integer translates f(x) and f(x+k) of a monic cubic polynomial over the integers. We derive an explicit formula for their resultant and use it to classify the possible degrees of their greatest common divisor after reduction modulo a prime. We then specialize to P(x) = x^3 - x^2 + 1, obtaining a shift-counting criterion and analyzing the associated cubic polynomial and its Galois group. The resulting conditions lead to a density statement for primes admitting nontrivial shifts.

Source: Zenodo (CERN European Organization for Nuclear Research)

DOI: 10.5281/zenodo.22135374

Author: Kumar, Aditya

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

OpenAlex ID: W7204565751

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