What type of study is this?

This is a Quantitative Study study.

October 13, 2025Open Access

On the height of polynomials that split completely over a fixed number field

Key Points

The minimal height of polynomials increases exponentially with degree n, confirming a significant result in number theory.
Exact implied constants were determined for specific fields, including totally real fields and certain quadratic extensions.
Polynomials of degree n show that roots in specific number fields lead to predictable height growth patterns.
This research contributes to understanding polynomial behavior in the context of finite extensions over the rationals.

Abstract

Let K/Q be a finite extension. We prove that the minimal height of polynomials of degree n of which all roots are in K^ increases exponentially in n. We determine the implied constant exactly for totally real K and K equal to Q (-1) or Q (-3).

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Cite This Study

Thian Tromp (Mon,) studied this question.

synapsesocial.com/papers/68ed1896f29694dd1da78af5 https://doi.org/https://doi.org/10.48550/arxiv.2509.12064

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