In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields. In the original case, the ultrametric field of interest was essentially the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ring K [ [ X ] ] {\displaystyle K[[X]]} , over K {\displaystyle K} , where K {\displaystyle K} was the real number or complex number field. This is still of considerable utility with respect to Puiseux expansions. The Newton polygon is an effective device for understanding the leading terms a X r {\displaystyle aX^{r}}
of the power series expansion solutions to equations P ( F ( X ) ) = 0 {\displaystyle P(F(X))=0}
where P {\displaystyle P} is a polynomial with coefficients in K [ X ] {\displaystyle K[X]} , the polynomial ring; that is, implicitly defined algebraic functions. The exponents r {\displaystyle r} here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in K [ [ Y ] ] {\displaystyle K[[Y]]}
with Y = X 1 / d {\displaystyle Y=X^{1/d}} for a denominator d {\displaystyle d} corresponding to the branch. The Newton polygon gives an effective, algorithmic approach to calculating d {\displaystyle d} . After the introduction of the p-adic numbers, it was shown that the Newton polygon is just as useful in questions of ramification for local fields, and hence in algebraic number theory. Newton polygons have also been useful in the study of elliptic curves.
Definition
A priori, given a polynomial over a field, the behaviour of the roots (assuming it has roots) will be unknown. Newton polygons provide one technique for the study of the behaviour of the roots. Let K {\displaystyle K} be a field endowed with a non-archimedean valuation v K : K → R ∪ { ∞ } {\displaystyle v_{K}:K\to \mathbb {R} \cup \{\infty \}} , and let
f ( x ) = a n x n + ⋯ + a 1 x + a 0 ∈ K [ x ] , {\displaystyle f(x)=a_{n}x^{n}+\cdots +a_{1}x+a_{0}\in K[x],}
with a 0 a n ≠ 0 {\displaystyle a_{0}a_{n}\neq 0} . Then the Newton polygon of f {\displaystyle f} is defined to be the lower boundary of the convex hull of the set of points P i = ( i , v K ( a i ) ) , {\displaystyle P_{i}=\left(i,v_{K}(a_{i})\right),} ignoring the points with a i = 0 {\displaystyle a_{i}=0} . Restated geometrically, plot all of these points Pi on the xy-plane. Let's assume that the points indices increase from left to right (P0 is the leftmost point, Pn is the rightmost point). Then, starting at P0, draw a ray straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point Pk1 (not necessarily P1). Break the ray here. Now draw a second ray from Pk1 straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point Pk2. Continue until the process reaches the point Pn; the resulting polygon (containing the points P0, Pk1, Pk2, ..., Pkm, Pn) is the Newton polygon. Another, perhaps more intuitive way to view this process is this : consider a rubber band surrounding all the points P0, ..., Pn. Stretch the band upwards, such that the band is stuck on its lower side by some of the points (the points act like nails, partially hammered into the xy plane). The vertices of the Newton polygon are exactly those points. For a neat diagram of this see Cassels 1986, chapter 6, §3.
Main theorem With the notations in the previous section, the main result concerning the Newton polygon is the following theorem, which states that the valuation of the roots of f {\displaystyle f} are entirely determined by its Newton polygon: Let μ 1 , μ 2 , … , μ r {\displaystyle \mu _{1},\mu _{2},\ldots ,\mu _{r}}
be the slopes of the line segments of the Newton polygon of f ( x ) {\displaystyle f(x)} (as defined above) arranged in increasing order, and let
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