Newton polygon

From Academic Kids

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

aXr

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, implicitly defined algebraic functions. The exponents r here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in

K[[Y]]

with Y = X1/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 <math>K<math> be a local field with discrete valuation function <math>v_K<math> and let

<math>f(x) = a_nx^n + \ldots+a_1x+a_0 \in K[x]<math>

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

<math>P_i=\left(i,v_K(a_i)\right)<math>

In non-jargon plot all of these points on a graph, then starting at <math>P_0<math> 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 <math>P_n<math>; this graph is the Newton polygon.

Applications

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

Let

<math>\mu_1, \mu_2, \ldots, \mu_r<math>

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

<math>\lambda_1, \lambda_2, \ldots, \lambda_r<math>

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 <math>P_i<math> and <math>P_j<math> then the length is <math>j-i<math>). Then for each <math>1\leq\kappa\leq r<math>, <math>f(x)<math> has exactly <math>\lambda_{\kappa}<math> roots with valuation <math>\mu_k<math>.

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