Questions tagged [transcendence-theory]

A brief study of transcendental numbers and algebraic independence theories; currently an on-development branch of mathematics with a lot of open problems.

Primarily, questions related to determination of transcendental numbers, theory of algebraic independence and diophantine approximation should have this tag. Closed form-related question can have this tag if is broad enough. Example: is $\text{K} \cap \mathbb{Q} = \lbrace\emptyset\rbrace$ for the ring $\text{K} \in \mathbb{C}$ defined as $(...)$?

Also known as transcendental number theory.

119 questions
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Existence of $x$ such that $2^x =a,3^x=b,5^x=c$ for some integers $a,b,c$

Conjecture: There does not exist a non-integer $x$ such that $$2^x=a$$ $$3^x=b$$ $$5^x=c$$ where $a,b,c$ are all integers. I'm aware that the similar question There does not exist a non-integer $y$ such that $$2^y=A$$ $$3^y=B$$ where $A,B$ are…
mick
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Transcendence Degree of the Function Field of Meromorphic Functions over $\mathbb{C}$

What is the transcendence degree of the field of meromorphic functions over $\mathbb{C}$? By a cardinality argument (meromorphic functions are determined by their image under a countable dense subset and $\mathbb{C}^{\mathbb{Q}}$ has the same…
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Are there classifications transcendental numbers that are similar to algebraic numbers for differential equations?

Considering that transcendental numbers are described as not a root of a non-zero polynomial equation with rational coefficients, are there classifications of transcendental numbers that are considered not algebraic but the solution to a definite…
dezakin
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irrationality measure

I was reading that you can associate a measure to any given number giving you "how irrational" the given number is. I was wondering is there any irrationality measure that would tell you that the number under consideration is 100 percent irrational.…
Adam
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Is there any $p \in \mathbb{Q}[x,y]$ with x and y degree both at least 1 such that $p(\pi,e)$ is known to be irrational?

The algebraic independence of $\pi$ and $e$ is a well known open problem, as is the specific case of rationality of $\pi + e$. My question is if there is any polynomial with rational coefficients (other than the obvious ones where e.g.…
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Field Extensions with Common Transcendental Numbers

Does anyone know if there is work done in this direction where one extends (the field) $\mathbb{Q}$ or $\bar{\mathbb{Q}}$ with certain common transcendental numbers such as $\pi$, $e$, etc. For example, can one "get away" with such a field extension…
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Non-zero coefficient in a transcendence proof

I am studying the proof of the simple version of Lindemann's Theorem. Theorem. If $\alpha$ is a non-zero algebraic number, then $e^\alpha$ is transcendental. Apologies for the long read, but to explain my question I need to give a very brief…
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What makes Hilbert's 7th problem important/relevant?

What was the motivation behind Hilbert's 7th problem? Looking into some of the history behind transcendental number theory, it seems that the field was almost non-existent in the late 1800's/early 1900's. There was the result of Liouville proving…
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Is there a proof that there is no general method to solve transcendental equations?

Being motivated by this post, I was wondering if there is a proof (analogous to the case of Diophantine equations) that there is no general method for solving transcendental equations? It seems pretty clear, intuitively, that there can be no…
Matt Calhoun
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Showing that Liouville's numbers are transcendental (Liouville's theorem)

From Vladimir Zorich Analysis I: Let us call an irrational number $a \in \mathbb{R}$ well approximated by rational numbers if for any natural number $n, N \in \mathbb{N}$ there exists a rational number $\frac{p}{q}$ such that $\lvert a -…
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What does algebraic independence mean?

If you search for algebraic independence, you will find the following on Wikipedia: In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non-trivial polynomial equation…
Kinheadpump
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Liouville-Roth Irrationality Measure of $\pi$ = 2 already proven?

I was looking through this paper and I was curious. Has it already been established that the Flint Hills series $\displaystyle\sum_{n\in\mathbb{Z}^{+}}\frac{\csc^2 n}{n^3}$ converges? And has it already been established that the Liouville-Roth…
El Ectric
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Rigidity of logarithms of positive integers

The following question is from my colleague. It seems to be emerged considering some elementary number theory problem. This is NOT as exercise or problem in published material although it may looks like. Consider the following equation of positive…
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Flat $\mathbb{Z}$-modules and algebraic independence

I know that this question is naive, but I really want to find out if this observation is correct. First, let $\alpha_1, \alpha_2, \dotsc, \alpha_k$ be real numbers. Consider the $\mathbb{Z}$-module homomorphism \begin{align*} f \colon…
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If the Liouville Constant is transcendental, is its exponentiation also transcendental?

So we have Liouville's Constant: $L_b=\displaystyle\sum_{n\in\mathbb{Z}^+}b^{-n!}=\left(0.\textbf{1}\textbf{1}000\textbf{1}00000000000000000\textbf{1}00000000...\right)_b$ And let $M$ be the constant defined as: (is there a name for this…
El Ectric
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