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joaopa
joaopa
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Let $\beta$ be an element of $\overline{\mathbb F_q(T)}\setminus\overline{\mathbb F_q}$. Is it true that the sequence $(\beta^{q^n}-T)_n$ admits infinitely many zeros, that is there exist infinitely many distinct places $\mathcal P_1,\cdots,\mathcal P_n\cdots,$ of $\mathbb F_(T)(\beta)$$\mathbb F_q(T)(\beta)$ such that for every $n\in\mathbb N$, there exists a $k\in\mathbb N$ such that $v_{\mathcal P_n}(\beta^{q^k}-T)>0$, where $v_{\mathcal P_n}$ is the valuation of $K$ associated to the place $\mathcal P_n$. That is an analogue of the following number theory problem: Let $\alpha,\beta\ge2$ be an integer. Does the sequence $(\alpha^n-\beta)_n$ admit infinitely many divisors?

Let $\beta$ be an element of $\overline{\mathbb F_q(T)}\setminus\overline{\mathbb F_q}$. Is it true that the sequence $(\beta^{q^n}-T)_n$ admits infinitely many zeros, that is there exist infinitely many distinct places $\mathcal P_1,\cdots,\mathcal P_n\cdots,$ of $\mathbb F_(T)(\beta)$ such that for every $n\in\mathbb N$, there exists a $k\in\mathbb N$ such that $v_{\mathcal P_n}(\beta^{q^k}-T)>0$, where $v_{\mathcal P_n}$ is the valuation of $K$ associated to the place $\mathcal P_n$. That is an analogue of the following number theory problem: Let $\alpha,\beta\ge2$ be an integer. Does the sequence $(\alpha^n-\beta)_n$ admit infinitely many divisors?

Let $\beta$ be an element of $\overline{\mathbb F_q(T)}\setminus\overline{\mathbb F_q}$. Is it true that the sequence $(\beta^{q^n}-T)_n$ admits infinitely many zeros, that is there exist infinitely many distinct places $\mathcal P_1,\cdots,\mathcal P_n\cdots,$ of $\mathbb F_q(T)(\beta)$ such that for every $n\in\mathbb N$, there exists a $k\in\mathbb N$ such that $v_{\mathcal P_n}(\beta^{q^k}-T)>0$, where $v_{\mathcal P_n}$ is the valuation of $K$ associated to the place $\mathcal P_n$. That is an analogue of the following number theory problem: Let $\alpha,\beta\ge2$ be an integer. Does the sequence $(\alpha^n-\beta)_n$ admit infinitely many divisors?

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joaopa
joaopa
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Zeros of a sequence in $\overline{\mathbb F_q(T)}$

Let $\beta$ be an element of $\overline{\mathbb F_q(T)}\setminus\overline{\mathbb F_q}$. Is it true that the sequence $(\beta^{q^n}-T)_n$ admits infinitely many zeros, that is there exist infinitely many distinct places $\mathcal P_1,\cdots,\mathcal P_n\cdots,$ of $\mathbb F_(T)(\beta)$ such that for every $n\in\mathbb N$, there exists a $k\in\mathbb N$ such that $v_{\mathcal P_n}(\beta^{q^k}-T)>0$, where $v_{\mathcal P_n}$ is the valuation of $K$ associated to the place $\mathcal P_n$. That is an analogue of the following number theory problem: Let $\alpha,\beta\ge2$ be an integer. Does the sequence $(\alpha^n-\beta)_n$ admit infinitely many divisors?