On the ranks and implicit constant fields of valuations induced by pseudo monotone sequences

Arpan Dutta

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arXiv · 2107.10570

On the ranks and implicit constant fields of valuations induced by pseudo monotone sequences

Arpan Dutta

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Abstract

Given a valued field $(K,v)$ and a pseudo monotone sequence $E$ in $(K,v)$, one has an induced valuation $v_E$ extending $v$ to $K(X)$. After fixing an extension of $v_E$ to a fixed algebraic closure $\overline{K(X)}$ of $K(X)$, we show that the implicit constant field of the extension $(K(X)|K,v_E)$ is simply the henselization of $(K,v)$. We consider the question: given a value transcendental extension $w$ of $v$ to $K(X)$ and a pseudo monotone sequence $E$ in $(K,v)$, under which precise conditions is $w$ induced by $E$? The dual nature of pseudo convergent sequences of algebraic type and pseudo divergent sequences is also explored. Further, we provide a complete description of the various possibilities of the rank of the valuation $v_E$, provided that $v$ has finite rank.

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Arpan Dutta. 2021-07-22. On the ranks and implicit constant fields of valuations induced by pseudo monotone sequences. https://arxiv.org/abs/2107.10570

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