Newton polygon

In mathematics , the Newton polygon is a tool for studying polynomials.

definition

The Newton polygon of is the convex hull of (2,3), (1,2) and (3,1), the point (2,2) lies inside. ${\displaystyle P(x,y)=}$ ${\displaystyle 3x^{2}y^{3}-xy^{2}+2x^{2}y^{2}-x^{3}y}$

Be it

${\displaystyle P(x,y)=\sum _{i,j}a_{ij}x^{i}y^{j}}$

a polynomial in two variables x and y. Then the Newton polygon of P is the convex hull of

${\displaystyle \left\{(i,j)\in \mathbb {R} ^{2}:a_{ij}\not =0\right\}}$.

Similarly, one can define a Newtonian polytope for polynomials in more than two variables. The definition can be generalized to polynomials with coefficients in an evaluated body.

Applications

Newton's polygons are used in analysis when solving non-linear systems of equations, in number theory when factoring polynomials over local fields and in topology when constructing node invariants.

literature

• Shui-Nee Chow, Jack K. Hale: Methods of Bifurcation Theory . (= Basic Teachings of Mathematical Sciences. Volume 251). Springer, New York 1982, ISBN 1-4613-8161-4 .
• Jürgen Neukirch: Algebraic number theory. Springer, Berlin 1992, ISBN 3-540-37547-3 .