Newton polygon
![Construction of the Newton polygon of the polynomial P(X) = 1 + 5 X + 1/5 X^2 + 35 X^3 + 25 X^5 + 625 X^6.](/Images/godic/202501/29/Newton-polygon0541.gif")
![This diagram shows the Newton polygon for P(x,y) = 3 x^2 y^3 - x y^2 + 2 x^2 y^2 - x^3 y, with positive monomials in red and negative monomials in cyan. Faces are labelled with the limiting terms they correspond to.](/Images/godic/202501/29/Diagram_of_a_Newton_Polygon_Convex_hull.svg0542.png")
In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields.
In the original case, the local field of interest was the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ring