Newton polygon

In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields. In the original case, which is still of considerable utility with respect to Puiseux expansions, the local field would be K X, the field of formal power series over K , which was the real number or complex number field, in the indeterminate X . In this case the Newton polygon is an effective device for understanding the leading terms

: aX r

of the power series expansion solutions to equations

: P ( F ( X )) = 0

where P is a polynomial with coefficients in K [ X ], the polynomial ring; that is, implicit function algebraic functions. The exponents r here are certain rational numbers, depending on the branch of a function chosen; and the solutions themselves are power series in

: K Y

with Y = X 1/ d for a denominator d corresponding to the branch. The Newton polygon gives an effective, algorithmic approach to calculating 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.

=Definition=

Let K be a local field with discrete valuation function (mathematics) v_K and let

:f(x) = a_nx^n + ldots+a_1x+a_0 in K[x]

Then the Newton polygon of f is defined to be the convex hull of the set of points

:P_i=left(i,v_K(a_i) ight)

In non-jargon plot all of these points on a graph, then starting at P_0 draw an imaginary line straight up parallel with the y-axis, rotate this line counter-clockwise until you hit a point, break the line here and keep rotating until you hit another... continue until you reach the point P_n; this graph is the Newton polygon.

= Applications =

The practical purpose of the Newton polygon comes from the following result:

Let

:mu_1, mu_2, ldots, mu_r

be the slopes of the line segments of the Newton polygon of f(x) (as defined above) arranged in increasing order, and let

:lambda_1, lambda_2, ldots, lambda_r

be the corresponding lengths of the line segments projected onto the x-axis (i.e. if we have a line segment stretching between the points P_i and P_j then the length is j-i). Then for each 1leqkappaleq r, f(x) has exactly lambda_{kappa} roots with valuation mu_k.