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.
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Is there a geometric theory of transcendental equations?
As you know, transcendental functions can be represented by infinite series expansions. Which means that transcendental equations can be expressed as polynomials of (countably)infinite degree. The ...
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Is this subclass of transcendental function studied in the literature?
By definition, an analytic function $f(x)$ is an algebraic function if there exists a rational non-zero polynomial $P(x,f(x))$ such that $P(x,f(x)) = 0$. If $f(x)$ is not transcendental, then $f(x)$ ...
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Images of a vector under the Galois differential group span the solution set
I am reading the paper "A refined version of the Siegel-Shidlovskii theorem" by F. Beukers. In the proof of Theorem 1.5, he mentions the following results in Galois differential theory. Let ...
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Surjective maps between algebraic groups induce surjective maps between connected components
I am reading the paper of F. Beukers, A refined version of the Siegel-Shidlovskii theorem (here is the link). The author mentions the following result in Algebraic group theory without proof.
Lemma 2....
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Does Schanuel's conjecture imply that $\pi^e$ is transcendental?
My understanding (and correct me if I'm wrong) is that it is unknown whether $\pi^e$ is algebraic or transcendental. I've also been led to believe that most open questions of this type would be solved ...
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Deduction from Schanuel's conjecture
I am deducing from the Schanuel's conjecture the following statement:
Let $\alpha, \beta$ be positive real algebraic numbers with $\alpha,\beta \neq 1$ and $\frac{\log \beta}{\log \alpha} \notin \...
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Transcendence of meromorphic function vs formal power series
Consider the meromorphic function $f$ on $\mathscr D=\{z\in\mathbb C\mid|z|<1\}$ definied by $\displaystyle f(z)=\sum_{n\ge1}\frac{z^{2^n}}{z^{2^n}-\frac12}$. Obviously $f$ admits infinitely many ...
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House of the inverse
Let $\alpha,\beta\in\overline{\mathbb Q}$. Denote by $h(\alpha)$ the house $\alpha$, that is the maximum of $|\sigma(\alpha)|$ when $\sigma$ describes $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$. ...
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Transcendence of function and change of fields
Suppose the one has a sequence of rational functions $Q_n(z)\in\mathbb Q(z)$. Let $p$ be a prime number. Suppose that that there exists an infinite subset $X$ of $\mathbb Q_p$ such that:
the ...
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What is $\text{trdeg}(F( X_i Y_j \mid 1 \leq i \leq n, 1 \leq j \leq m) / F)$?
We have indeterminate variables $X_i$ and $Y_j$ for $1 \leq i \leq n$ and $1 \leq j \leq m$.
It is known $\text{trdeg}(L/F) = \text{trdeg}(L/K) + \text{trdeg}(K/F)$ for every field extension $F \...
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Transcendence measure for the canonical Liouville number
Let $\displaystyle\alpha=\sum_{n=0}^{+\infty}\frac1{10^{n!}}$. It is well-known that $\alpha$ is transcendental. I am looking for a transcendental measure for $\alpha$. That is exercise 11.15 of the ...
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Let $L = F(S)$ for $S \subseteq L$. Does there exists a transcendental basis $T \subseteq S$ with $F(S) = F(T)$?
Let $L/F$ be a field extension with $L = F(S)$ for a subset $S \subseteq L$. My first question is:
Is there a subset $T \subseteq S$ such that $T$ is a transcendental basis?
Now let's say we have $I ...
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Real number known not to be a period
I am working a bit with problems in non-archimedean settings inspired by the famous periods conjecture by Kontsevich-Zagier. I was preparing a talk and wanted to give of background about the initial ...
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Is the statement of Baker's Theorem on wikipedia inaccurate?
The Wikipedia article on Baker's theorem states it as follows:
If $\lambda_1,\ldots,\lambda_n\in\mathbb{L}$ are linearly independent over the rational numbers, then for any algebraic numbers $\beta_0,...
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Show, using a specific approach, that $\dim \Bbb P^n=\dim\Bbb A^n=n$.
This is Exercise 1.8.4(1) of Springer's, "Linear Algebraic Groups (Second Edition)". It is not a duplicate of
The dimension of $\mathbb P^n$ is $n$
because I'm after a particular perspective;...
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