January | 2019 | Hydrobates

Archive for January, 2019

Minimal introduction to Newton polygons

January 24, 2019

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.