Newton polygon

Summary

The Newton polygon of a polynomial over a valued field is a broken line built from its coefficients $a_i$, see the definition. It is the lower boundary of the convex hull of the points $(i, v(a_i))$. Its slopes and their lengths give the valuations of the roots and how many roots have each valuation. It is a quick tool for factoring polynomials over $p$-adic fields and for finding Puiseux series solutions of algebraic equations.