Minimal introduction to Newton polygons

In my work on dynamical systems I have used Newton polygons as a practical tool but I never understood the theoretical basis for why they are helpful. Here I give a minimal theoretical discussion. I do not yet understand the link to the applications I just mentioned but at least it is a start. Consider a polynomial equation of the form p(x,y)=0. The polynomial p can be written in the form p(x,y)=\sum_{i,j}a_{ij}x^iy^j. Suppose that p(0,0)=0, i.e. that a_{00}=0. I look for a family y=u(x) of solutions satisfying u(x)=Ax^\alpha+\ldots. We have F(x,u(x))=0. Intuitively the zero set of p is an algebraic variety which near the origin is the union of a finite number of branches. The aim is to get an analytic approximation to these branches. Substituting the ansatz into the equation gives a_{ij}x^iy^j=a_{ij}A^jx^{\alpha j+i}+\ldots=0. If we compare the size of the summands in this expression then we see that summands have the same order of magnitude if they satisfy the equation \alpha j+i=C for the same constant C. Let S be the subset of the plane with coordinates (i,j) for those cases where a_{ij}\ne 0. For C=0 the line L with equation \alpha j+i=C does not intersect S. If we increase C then eventually the line L will meet S. If it meets S in exactly one point then the ansatz is not consistent. A given value of \alpha is only possible if the line meets S in more than point for some C. Let \tilde S be the set of points with coordinates (k,l) such that k\ge i and l\ge j for some (i,j)\in S and let K be the convex hull of \tilde S. Then for an acceptable value of \alpha the line L must have a segment in common with K. There are only finitely many values of \alpha for which this is the case. A case which could be of particular interest is that of the smallest branch, i.e. that for which \alpha takes the smallest value. Consider for simplicity the case that only two points of L belong to S. Call their coordinates (i_1,j_1) and (i_2,j_2). Then the coefficient A is determined by the relation A^{j_2-j_1}=-\frac{a_{i_1j_1}}{a_{i_2j_2}}. Further questions which arise are whether the formal asymptotic expansion whose leading term has been calculated can be extended to higher order and whether there is a theorem asserting the existence of a branch for which this is actually an asymptotic approximation. These will not be pursued here.

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