#
About Cogredient and Contragredient Linear Differential Equations^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

- (1)
- all the numbers ${c}_{n}$ belong to an algebraic field $\mathbb{K}$ of finite degree over $\mathbb{Q}$;
- (2)
- for arbitrary $\epsilon >0\phantom{\rule{0.277778em}{0ex}}$$\phantom{\rule{0.277778em}{0ex}}\overline{|{c}_{n}|}=O\left({n}^{\epsilon n}\right),\phantom{\rule{1.em}{0ex}}n\to \infty $, where $\overline{\left|\alpha \right|}$ is the maximum of the absolute values of the algebraic number $\alpha $ and all its conjugates in the field $\mathbb{K}$;
- (3)
- for arbitrary $\epsilon >0$ the least common denominator of ${c}_{1},\cdots ,{c}_{n}$ is $O\left({n}^{\epsilon n}\right),\phantom{\rule{1.em}{0ex}}n\to \infty .$

**Example**

**1.**

**Theorem**

**1.**

## 2. Proof of the Theorem 1

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Proof of Lemma 3.**

**Lemma**

**4.**

**Corollary**

**1.**

**Proof of Lemma 4.**

**Lemma**

**5.**

**Corollary**

**2.**

**Proof of Lemma 5.**

**Proof of Theorem 1.**

## 3. Conclusions

**Example**

**2.**

## Funding

## Conflicts of Interest

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Gorelov, V.
About Cogredient and Contragredient Linear Differential Equations. *Axioms* **2021**, *10*, 117.
https://doi.org/10.3390/axioms10020117

**AMA Style**

Gorelov V.
About Cogredient and Contragredient Linear Differential Equations. *Axioms*. 2021; 10(2):117.
https://doi.org/10.3390/axioms10020117

**Chicago/Turabian Style**

Gorelov, Vasily.
2021. "About Cogredient and Contragredient Linear Differential Equations" *Axioms* 10, no. 2: 117.
https://doi.org/10.3390/axioms10020117