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Questions tagged [transcendence]

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5
votes
1answer
200 views

$f,g \in \mathbb{Z}[x,y]$ satisfying: $\operatorname{Jac}(f,g)=0$ and $f,g \notin \mathbb{Z}[h]$ for every $h \in \mathbb{Z}[x,y]$?

Is it possible to find $f,g \in \mathbb{Z}[x,y]$ (with $\deg(f),\deg(g) \geq 1$) such that the following two conditions are satisfied: (1) $\operatorname{Jac}(f,g)=f_xg_y-f_yg_x = 0$. (2) ...
5
votes
1answer
324 views

Number defined by a recursive binary sequence

In a math column in Scientific American many years ago, I encountered a peculiar binary sequence I describe below. Unfortunately I can't find a reference on this, so I would be grateful for any ...
5
votes
0answers
163 views

Transcendental Continued Fractions

Liouville famously showed the existence of a transcendental number by considering $\alpha =\sum\limits_{n=0}^\infty10^{-n!}$ and showing that it did not satisfy 'Liouville's criterion' $|\alpha-p/q|\...
6
votes
1answer
208 views

If $f,g \in D[x,y]$ are algebraically dependent over $D$, then $f,g \in D[h]$ for some $h\in D[x,y]$?

This question asks: If $f,g \in k[x,y]$ are two algebraically dependent polynomials over an arbitrary field $k$, is it true that there exists a polynomial $h \in k[x,y]$ such that $f,g \in k[h]$, ...
5
votes
2answers
178 views

A non-commutative analog of: two polynomials are algebraically dependent iff their Jacobian is zero

Let $f,g \in \mathbb{C}[x,y]$. There is a well-known result, that can be found for example here, pages 19-20, that says the following: $f,g$ are algebraically dependent over $\mathbb{C}$ if and ...
2
votes
0answers
230 views

Generalizations of Lüroth theorem

Let $k$ be an arbitrary field (I do not mind to take $k=\mathbb{C}$, if things are easier in this case). A more general version of Lüroth theorem says that a field $L$, $k \subset L \subset k(x,y)$, ...
6
votes
1answer
215 views

Transcendence degree of the fraction field of $k[G]$ for torsion free abelian group $G$

Let $k$ be a field of characteristic $p$ and $G$ be a torsion free abelian group . Then the group ring $k[G]$ is an integral domain , let $k(G)$ denote its field of fractions . Then can we say ...
1
vote
0answers
108 views

Nielsen--Schreier for fields

Is it true that a subextension of a purely transcendental extension is itself purely transcendental? In symbols, suppose we have field extensions $M/L/K$, with $M/K$ purely transcendental. Must $L/K$ ...
7
votes
1answer
748 views

Why is the Euler-Mascheroni constant not a Liouville number?

Let $\gamma$ be the Euler-Mascheroni constant. Why is $\gamma$ not a Liouville number? Are there any upper bounds for the irrationality measure of $\gamma$ known? Any pointers to the literature are ...
11
votes
1answer
390 views

“Transcendental tilings”: Do they exist?

Let $T$ be a tiling of the plane. Fix an origin and shoot a ray $r$ from the origin. Mark off points $p_i$ along $r$ separated by unit distance. Compute from $r$ a binary number $0 < b(r) < 1$ ...
2
votes
1answer
214 views

Solving a transcendental equation, in closed form

There is a change of variable between two varibles $q$ and $Q$ as the following: $$q=Q\exp(2f(Q))\quad\quad \quad (*)$$ where $f(Q)$ is given by $$f(Q)=\sum_{d=1}^\infty \frac{(2d-1)!}{(d!)^2}Q^d$$ ...
6
votes
1answer
474 views

Transcendence of $e^{\frac{\pi^2}{12 \log 2}}$

Is it known whether $e^{\frac{\pi^2}{12 \log 2}}$ is transcendental or algebraic? This number showed up in this other question.
3
votes
0answers
126 views

$\mathbb{Z}$-linear independence of arguments of units in non-CM number fields

Playing a little bit with Groessencharacters a stumbled on the following question: Let $K$ be a non CM number field with $r_1$ real embeddings and $2r_2>0$ complex embeddings. Set $r=r_1+r_2-1$ ...
8
votes
0answers
236 views

Transcendence of the $p$-adic number $\sum_{n\ge0}a^{2^n}$

Let $p$ a prime number and $a\in\overline{\mathbb Q}\subset\mathbb C_p$ be an algebraic $p$-adic number such that $|a|_p<1$. Is the number $s_a=\sum_{n\ge0}a^{2^n}$ transcendental over $\mathbb ...
-1
votes
1answer
73 views

Preservation of algebraically dependence for derivative [closed]

It is well-known (see Allouche monography for example) that if $f$ is an algebraic function over $K(X)$ then $f'$ is also algebraic. I wonder whether $f$ and $g$ are algebraically dependent, then $f'$ ...
0
votes
2answers
285 views

algebraic independence of exponential functions

Let $a_1,\ldots,a_n$ be $\mathbb Q$-linearly independant algebraic numbers. Are the functions $e^{az},\ldots,e^{a_nz}$ algebraically independent functions (over $\mathbb C(z)$ or $\mathbb Q(z)$)? I ...
2
votes
1answer
234 views

Number of generators of ideal if quotient field has certain transcendence degree

I am trying to prove a result which is used in A. Macintyre and A. J. Wilkie (1995), 'On the decidability of the real exponential field', in Odifreddi, P.G., Kreisel 70th Birthday Volume, CLSI, p. ...
0
votes
1answer
129 views

Transcendence of solutions of $\sum_{i=1}^n a_i b_i^x = 1$

I am currently doing things related to the Akra–Bazzi theorem. One element in that theorem is the following: For $n>0$ and sequences of real numbers $a_i, b_i$ of length $n$, where all $a_i>0$ ...
12
votes
1answer
517 views

Are there integral representations of the Mertens constant?

It is well-known that the Euler constant $$\gamma=\lim\limits_{n\to \infty}\biggl( \sum\limits_{k\le n}\dfrac{1}{k}-\ln{n}\biggr ) $$ has a bunch of integral representations, e.g. $$\gamma=-\int\...
11
votes
1answer
446 views

Transcendence of products of certain real algebraic numbers

Let \begin{equation} z := \prod_p p^{1/p^2}, \end{equation} where the product is over all prime numbers $p$, and we always take the positive real root. Is $z$ transcendental or algebraic, or (as I ...
4
votes
1answer
214 views

Hermite Lindemann and transcendental reals

This is about compass and straightedge constructions, although I suspect nothing changes if we expand to real numbers algebraic over the rationals. Anyway, let $E$ be the "constructible numbers," ...
4
votes
3answers
342 views

Algebraically Independent Numbers and Affine Linear Maps

Suppose I have two lists $\alpha_1,\dots,\alpha_k$ and $\beta_1,\dots,\beta_k$ of real numbers such that all $2k$ numbers are mutually algebraically-independent over the rationals. For each $i \in \{1,...
17
votes
2answers
750 views

Numbers with known finite irrationality measure greater than 2

For a real number $\alpha$, let the irrationality measure $\mu(\alpha) \in \mathbb{R}\cup \{\infty\}$ be defined as the supremum of all real numbers $\mu$ such that $$ \left| \alpha-\frac{p}{q}\right|...
-2
votes
1answer
344 views

Random Sequence : Definition of [closed]

"A sequence of bits is random if there exists no Program shorter than it which can produce the same sequence." ~ Kolmogorov Q: How do the digits of Pi fall as a random sequence based on the above ...
33
votes
2answers
9k views

Why is it hard to prove that the Euler Mascheroni constant is irrational?

Philosophically why should proving that $\gamma$ is irrational (let alone transcendental) be so much harder than proving $\pi$ or $e$ are irrational?
5
votes
2answers
891 views

What was Lambert's solution to $x^m+x=q$?

I've been thinking about Lambert's Trinomial Equation quite a bit, and I want to see his solution. The only solution I could find was in Euler's form, and I still don't quite understand how he got ...
3
votes
0answers
389 views

quasi periodic continued fractions and powers of e, tanh, tan

It is well known that some transcendental numbers like e.g. rational multiples of $e^{2/n}$ with $n\in\mathbb N $, when written as regular continued fractions (R.C.F.), yield what can be called a ...
4
votes
0answers
292 views

Transcendental numbers that are “suspected” to be algebraically dependent without conjectured relation?

I am experimenting with a solver for finding algebraic dependencies and would like to test it on more data sets. Are there transcendental numbers that are "suspected" to be algebraically dependent ...
24
votes
1answer
2k views

Is this seemingly novel irrational constant also transcendental?

I recently discovered a constant that is constructed as follows: $\chi=\sum_{n=1}^\infty (\frac{\cos{n}}{2|\cos{n}|}+\frac{1}{2}) 2^{-n}$ Furthermore I can prove that it is an irrational number ...
4
votes
1answer
348 views

Finding purely transcendental parts of field extensions

If we have a field $K$ such that $K\cong K(t)$ (i.e. it is isomorphic to the field you get if you adjoin one transcendental) then is there necessarily a subfield $L\lt K$ such that $L\ncong L(t)$ and $...
10
votes
0answers
565 views

Connections of results in Harmonic analysis in the theory of Transcendental Numbers

An entire function $f$ is said to be of exponential type if there exist constants $c$ and $k$ such that $|f(z)|\leq c e^{k |z|}$. A famous result of Polya says if $f$ is an entire function of ...
10
votes
5answers
1k views

Euler's divergent series sum n!*(-1)^n: what is known about the resulting constant?

Much of the theory of continued fractions has been developped by Euler in the 18th century. The little survey "Euler: continued fractions and divergent series (and Nicholas Bernoulli)", mentions ...
7
votes
2answers
924 views

Is this number already known to be transcendental? Is there a survey about up-to-date trascendence results?

Is the number $\sum_{n=1}^\infty \frac{1}{2^{n^2}}$ known to be transcendental? Is there a survey with up-to-date transcendence results?
1
vote
0answers
316 views

Extending transcendental numbers to multivariate polynomials [closed]

Transcendental numbers are a well-known phenomenon: a number $x$ is transcendental if no polynomial with integer coefficients has $x$ as its root, or $p(x)\neq0$ for all polynomials $p$ with integer ...
36
votes
6answers
4k views

What is the simplest, most elementary proof that a particular number is transcendental?

I teach, among many other things, a class of wonderful and inquisitive 7th graders. We've recently been studying and discussing various number systems (N, Z, Q, R, C, algebraic numbers, and even ...
21
votes
2answers
2k views

Have all numbers with “sufficiently many zeros” been proven transcendental?

Any number less than 1 can be expressed in base g as $\sum _{k=1}^\infty {\frac {D_k}{g^k}}$, where $D_k$ is the value of the $k^{th}$ digit. If we were interested in only the non-zero digits of this ...
24
votes
3answers
4k views

Work on independence of pi and e

It is an open problem to prove that $\pi$ and $e$ are algebraically independent over $\mathbb{Q}$. What are some of the important results leading toward proving this? What are the most promising ...
16
votes
3answers
1k views

Striking applications of Baker's theorem

I saw that there are many "applications" questions in Mathoverflow; so hopefully this is an appropriate question. I was rather surprised that there were only five questions at Mathoverflow so far with ...
10
votes
1answer
539 views

Are periods of rigid Calabi-Yau threefolds over $Q$ algebraic?

Let $X$ be a (smooth) compact complex manifold, and suppose that $H^1(X, \Theta_X) = 0$, where $\Theta_X$ is the tangent sheaf. In other words, suppose that $X$ is rigid. Suppose moreover that $X$ ...
19
votes
6answers
4k views

Showing e is transcendental using its continued fraction expansion

Can the transcendence of e be shown using its continued fraction expansion e = [2;1,2,1,1,4,1,1,6,...]?
50
votes
4answers
11k views

Proof that pi is transcendental that doesn't use the infinitude of primes

I just taught the classical impossible constructions for the first time, and in finding my class a reference for the transcendence of pi, I found a dearth of distinct proofs. In particular, those ...