Questions tagged [transcendence]
The transcendence tag has no usage guidance.
61
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Exponential of Liouville Numbers
By Mahler classification of Transcendental real numbers (into the sets of $S$-, $T$- and $U$-numbers), we know that
Any Liouville number is a $U$-number.
$\log \alpha$ is either an $S$- or a $T$-...
0
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3
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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 ...
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Transcendence of functions and change of field of definition
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 ...
0
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Analytical proof of an equation that includes transcendental functions
Can anyone help me find an analytical proof for the following statement:
for
$$t \in ]0,1[\setminus\{t_0\},\quad t' \in ]0,1[\setminus\{t_0\},\quad t\neq t',\quad k_1,k_2 \in \mathbf{R}^*,\quad t_0 \...
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Siegel's method for transcendence measure quoted by Mahler
In "Zur Approximation der Exponentialfunktion und des Logarithmus. Teil II" Malher wrote (footnote 5 page 148) that the constant appearing in Satz 4 can be improved by a method communicated ...
4
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Transcendence measure: of $\ln(a/b)$
In the book " Number Theory IV Transcendental Numbers" written by Parsin and Shafarevich (book, page 104) it is asserted that to explicit a transcendence measure of a complex number $w$, it ...
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Liouville numbers with some "special" convergents
Recall that a real number $\ell$ is called a Liouville number, if there exists an infinite sequence of rational numbers $(p_n/q_n)_n$ for which
$$
0<\left|\ell-\frac{p_n}{q_n}\right|<\dfrac{1}{...
63
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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 ...
20
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Reciprocals of Fibonacci numbers
Is the sum of the reciprocals of Fibonacci numbers a transcendental?
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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?
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Ground state energy of anharmonic oscillator: algebraic or transcendental?
Consider the quantum anharmonic oscillator, with Hamiltonian $H=p^2/2+q^2/2+gq^4$ for some real $g\geq 0$, with $p$ and $q$ obeying the usual Heisenberg commutation relations. For $g=0$, the ground ...
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Irrationality or transcendence of $i^{i\Omega}$ and $2^\Omega$, with $\Omega=W(1)$ and $W(x)$ being the main branch of Lambert $W$ function
In this post we denote the main (or principal) branch of the Lambert $W$ function as $W(x)$, I add as reference that Wikipedia has the article Lambert $W$ function. The particular value $W(1)=\Omega$ ...
6
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Transcendence of values of Fredholm series at algebraic arguments
Let $d$ be an integer greater than $1$ and let $f(z)=\sum_{n\ge0}z^{d^n}$ be the Fredholm series. It is well-known that the Roth-Ridout theorem implies that $f(r)$ is a transcendental number for all ...
6
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Reference request for results that involve the transcendence degree
Currently I'm reading "On the Decidability of the Real Exponential Field" by Macintyre and Wilkie and the Proof of Theorem 1.1 (page 462-464) uses two algebraic results that involve the ...
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Transcendence on $ \alpha+f(\alpha), \alpha f(\alpha) $ and $ \alpha/f(\alpha) $ where $ \alpha$ is transcendental
Let $f(x)$ be a real transcendental function with algebraic coefficients. So $f(x)$ and $x$ are algebraically independent. Let $\alpha$ be a transcendental number, are the numbers $$\alpha+f(\alpha),\...
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Why is it easier to prove $e$ is transcendental than $\pi$?
Why is it easier to prove $e$ is transcendental than $\pi$?
I noticed that the proofs of $\pi$'s transcendence are much longer and have more details to check than those of $e.$ My guess is that it's ...
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How to prove that the solution to $x^{x+1}=(x+1)^{x}$ is transcendental?
I was asked by an high school student if there is an algebraic way to find the exact value of the solution to the equation
\begin{equation}\label{eq}
x^{x+1}=(x+1)^x
\end{equation}
Let us define that ...
3
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1
answer
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Can one define a degree of a period?
In the context of Feynman integrals, certain values of the zeta function appear at certain loop orders. Given that the tree-level amplitudes are rational, and assuming the zeta values to be ...
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Irrationality of $e^{x/y}$
How to prove the following continued fraction of $e^{x/y}$
$${\displaystyle e^{x/y}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+{\cfrac {x^{2}}{18y+\ddots }}}}}}}}}}...
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Transcendence à la Liouville
Let $\alpha\in\overline{\mathbb Q}^*$. How to prove that for a non ultimately constant sequence $(\varepsilon_n)_n$ with $\varepsilon_n\in\{0,1\}$, the number $\sum_{n\ge0}\frac{\varepsilon_n}{\alpha^{...
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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 ...
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Transcendence of Euler series
Is the Euler series $\sum_{n\ge0}n!X^n\in\mathbb C[[X]]$ transcendental over $\mathbb C(X)$?
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On a transcendental number defined as a variation involving the Lambert $W$ function in the nested square root representation of the golden ratio
Define the real number $\xi$ satisfying
$$\xi=\sqrt{1+W\left(1+\sqrt{1+W(1+\ldots)}\right)}\tag{1}$$
where $W(x)$ denotes the main branch of the Lambert $W$ function, as reference I add that Wikipedia ...
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Concerning $k \subset L \subset k(x,y)$
The following is a known result in algebraic geometry:
Let $k$ be an algebraically closed field of characteristic zero (for example, $k=\mathbb{C}$).
Let $L$ be a field such that $k \subset L \subset ...
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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|...
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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 ...
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$a^b=b^a$ and algebraicity [closed]
Suppose $a$ and $b$ are reals such that $a^b=b^a$. If $a$ is algebraic, is $b$ algebraic too?
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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 ...
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$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
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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 ...
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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|\...
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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]$,
...
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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 ...
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0
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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)$, ...
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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?
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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 ...
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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 ...
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0
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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$ ...
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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 ...
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"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
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1
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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
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1
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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
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$\mathbb{Z}$-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
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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 ...
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1
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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 $f'$ ...
2
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1
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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. ...
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1
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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 ...
0
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1
answer
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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 $a_i>0$ ...
4
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1
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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," ...
22
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2
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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 ...