The Newton polygon of a polynomial over a valued field is the lower convex hull of the points formed from coefficient indices and valuations. Its slopes determine the valuations of roots, counted with multiplicity.
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A Newton polygon is a geometric tool used in number theory and algebraic geometry, particularly in the study of polynomials and algebraic equations. It provides a way to analyze the behavior of a polynomial function at various points and helps in determining the properties of its roots, as well as understanding the multiplicity of these roots.