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5

No. $M\mathrel{:=}k(x,y)$ has a $k$-automorphism $\sigma:x\mapsto 1/x,\,y\mapsto 1/y$, of order 2. Let $G\mathrel{:=}\langle\sigma\rangle$, and put $L\mathrel{:=}M^{G}$, the fixed field. The elements $x+1/x$ and $y+1/y$ of $L$ are algebraically independent over $k$, hence $L$ has transcendency degree 2. However, no non-constant polynomial $g\in k[x,y]$ can ...

3

The irrationality measure of the Champernowne constant $C_b$ in base $b>2$ is exactly $b$.

2

A quick thought for your Question 2. If the limit for $\xi_f$ exists, then as you noted we'll get $\xi_f ^2 - 1 = f(\xi_f +1)$. If we define $g(x) = f(x+1) - x^2 +1 = f(x+1) - (x+1)^2 + 2(x+1)$, then we have $x^2 - 1 = f(x+1)$ iff $g(x) = 0$. So your second question more or less reduces to saying we have a function $g$, and we would like to know when the ...

2

This recent result Root separation for trinomials by Koiran also uses Baker's Theorem. In my PhD thesis, we use this result to demonstrate a polynomial time algorithm to isolate real roots of integer trinomials.

1

Regarding the expression of $f(1)$ in terms of other known constants: $$f(1) = -e\left(\gamma + \sum_{n\geq 1} (-1)^n \frac1{n\cdot n!} \right) \qquad\quad \tag{*} \label{id}$$ $$\qquad\qquad\qquad= -e \left(\gamma - 1 + \frac1{4} - \frac1{18} + \frac1{96} - \frac1{600} + \cdots \right)$$ where $e = \sum_{n\geq0} \frac1{n!} \approx 2.718$ and \$\gamma \...

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