Continuous morphism in function fields with extra conditions

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joaopa

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joaopa Asks: Continuous morphism in function fields with extra conditions
Let $\Omega$ be the completion of $\overline{\mathbb F_q\left(\left(\frac1T\right)\right)}$ for the topology induced by the valuation $-\deg$ on $\mathbb F_q\left(\left(\frac1T\right)\right)$. Consider $s$ distinct non zero elements $\alpha_1,\cdots,\alpha_s$ of $\overline{\mathbb F_q(T)}$. One assumes that $\deg(\alpha_1)\le\cdots\le\deg(\alpha_s)$. Does there exist a non zero morphism $\sigma$ of $\Omega$ such that $\sigma\left(\overline{\mathbb F_q(T)}\right)\subset\overline{\mathbb F_q(T)}$ and $\deg(\sigma(\alpha_i))<\deg(\sigma(\alpha_{i+1}))$ for any $i\in[1,s-1]$

Be careful for the sign. In the third line, it is $\le$ and the fourth one $<$.

Thanks in advance for any answer.

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screen capture of code and output

Code: 
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Tanmoy Pati Asks: Parallel transport in curved spacetime
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