# Bound for roots of a polynomial with coefficients in a non-Archimedean valued field

Is there any bound for the valuation of the roots of a given polynomial with coefficients in an algebraically closed non-Archimedean valued field? Any reference or insight would be appreciated.

• You are aware of the theory surrounding the Newton polygon? – Torsten Schoeneberg Apr 18 at 5:35
• @TorstenSchoeneberg, no I am not aware of it. Can you please recommend any reference to study and see how that is related to bounds for roots of a polynomial in the non-Archimedean case? – Chilote Apr 18 at 16:02
• The Newton polygon, which is taylormade for non-Archimedean settings, relates the valuations of a polynomial's roots to those of its coefficients (and I assume that is what you want, else I misunderstand the question). There is a succinct Wikipedia article, here are notes by Casselman: math.ubc.ca/~cass/research/pdf/Newton.pdf, and e.g. math.stackexchange.com/q/135451/96384, math.stackexchange.com/q/14148/96384 are posts here which discuss it. – Torsten Schoeneberg Apr 18 at 18:24
• I am working with Hanh fields and Levi-Civita fields (dense valuation and residue class field = the field of complex numbers) so very far from local fields. I am studying eigenvalues of some operator on $C_0$ over these type of fields. That's why I need some tools to study the roots of such polynomials. Thanks for the references, I will see what can be useful. – Chilote Apr 18 at 18:38
• You can check out the book Asymptotic Differential Algebra and Model Theory of Transseries which has a somewhat detailed section about valued fields and the Newton polygon. I expect there are better sources for this (maybe look for books of Ribenboim), but this is where I learned about this. Also, good to know about valued fields in general. – nombre Apr 18 at 20:00