Introduction

Alexander D. Bruno    Keldysh Institute of Applied Mathematics, Moscow

1 Concepts of Power Geometry

Many problems in mechanics, physics, biology, economics and other sciences are reduced to nonlinear equations or to systems of such equations. The equations may be algebraic, ordinary differential or partial differential; and systems may comprise the equations of one type, but may include equations of different types. The solutions to these equations and systems subdivide into regular and singular ones. Near a regular solution the implicit function theorem or its analogs are applicable, which gives a description of all neighboring solutions. Near a singular solution the implicit function theorem is inapplicable, and until recently there had been no general approach to analysis of solutions neighboring the singular one. Although different methods of such analysis were suggested for some special problems.

The purpose of the book is to supply a general purpose set of algorithms for analysis of singularities applicable to all types of equations. At present, the usual way of development of mathematical sciences may be depicted as the following sequence:

→ Theory → Algorithm →Software → Computation → Application,

where some elements may be in plural and feedbacks play a major part [Bruno 1998a]. This book comprises all elements of the sequence (1.1).

The main concept of Power Geometry is to study the properties of solutions to an equation through the power exponents of its monomials. For instance, to the polynomial

(X)=ΣfQXQ,Q∈S,

where X = (x1,…,xn), Q = (q1,…,qn), Q=x1q1…xnqn, there corresponds the set S ⊂ n of the vector power exponents Q, for which the coefficients fQ ≠ 0. Together with the set S (the support of Polynomial (1.2)), we consider its convex hull Γ (the Newton polyhedron of Polynomial (1.2)) and faces j(d) of the polyhedron Γ in the space n, and also their normal cones j(d) in the dual space *n that is the space of logarithms of coordinates xi. To each face j(d) there corresponds the truncation of Polynomial (1.2)

^j(d)(X)=ΣfQXQ,Q∈S∩Γj(d).

It is the first approximation to the Polynomial (1.2) in the part of X-space that corresponds to the normal cone j(d) in the (ln X) -space *n. In that part of the X-space, the first approximation to a solution to the equation f(X) = 0 is the solution to the truncated equation ^j(d)(X)=0. The power transformations of coordinates, which are linear transformations of logarithms of coordinates, induce the linear transformations in spaces n and *n, which allows to reduce the number of coordinates in a problem, especially for the truncated equations. The linear transformations in n and *n make Power Geometry a geometry in the Klein’s sense (see [Klein 1872]).

Example. Consider a plane (i.e. n = 2) algebraic curve, which is called the folium of Descartes and is defined by the equation

(X)=defx13+x23−3x1x2=0.

Let us study solutions to this equation near the origin x1 = x2 = 0 and at infinity, where the implicit function theorem can not be applied. The set S of power exponents of the equation consists of three points: Q1 = (3, 0), Q2 = (0, 3), Q3 = (1, 1). Their convex hull Γ is the triangle (Fig. 0.1, a). The points Q1 and Q3 lie in its edge 1(1). Hence, to it there corresponds the truncated equation

^1(1)(X)=defx13−3x1x2=0.

Its nontrivial solution 2=x12/3 is the first approximation to the branch 1 of solutions to Equation (1.4) (see Fig. 0.2). The branch 1 passes through the point x1 = x2 = 0. The points Q2 and Q3 lie on the edge 2(1); to it there corresponds the truncated equation 23−3x1x2=0. Its nontrivial solution 2=x12/3 the first approximation to the second branch 2 passing through the origin. To obtain subsequent approximations to the branch 1, let us make the power transformation

y1=x12x2−1,y2=x1−1x2,with inverse{x1=y1/y2,x2=y1y22.

Then Equations (1.4) and (1.5) are transformed into

=defy12y23(y1+y1y23−3)=0,f^1(1)=defy12y23(y1−3)=0.

Cancelling 12y23, we obtain the complete equation

1+y1y23−3=0

and its truncated equation y1 − 3 = 0. For Equation (1.8), the support and polyhedron are shown in Fig. 0.1, b. The root y1 = 3 of the truncated equation is simple. Hence in the neighborhood of the point y1 = 3,y2 = 0 the implicit function theorem is applicable to the complete Equation (1.8), which allows to obtain y1 − 3 as a power series of y2. Here it may be done explicitly: 1=3/(1+y23). Substituting this expression in right Formulae (6), we obtain the parametric representation for the branch 1:

1=3y2/(1+y23),x2=3y22/(1+y23).

The points Q1 and Q2 lie on the edge 3(1), and to it there corresponds the truncated equation 13+x23=0. Its only real solution x1 + x2 = 0 is the first approximation to the asymptote x1 + x2 = −1 of the folium of Descartes, shown in Fig. 0.2. To obtain the asymptotic expansion for its branches that go into infinity we need to apply the power transformation 1=x1,x2=x1−1x2.

Exercise. Plot the support and the Newton polyhedron for the left Equation (1.7).

2 Historical remarks

Power Geometry is based upon the three concepts: the Newton polyhedron, the power transformation and the logarithmic transformation. The crucial points of their development are as follows.

I. The Newton polyhedron. For n = 2, approximately in 1670 Newton [1711] suggested to use one edge of the “Newton open polygon” [Bruno 1979a] of a polynomial f(x, y) to find the branches of solutions to the equation f(x, y) = 0 near the origin x = y = 0, where the polynomial f has no constant or linear terms. Puiseux [1850] was already using all the edges of the Newton open polygon and had given a rigorous substantiation to the solution of the problem by this method. Liouville [1833] was using this approach to find the rational solutions y = y(x) to the linear ordinary differential equation

0(x)dny/dxn+…+an−1(x)dy/dx+an(x)=0,

where αi(x) are polynomials. Briot and Bouquet [1856] were using an analog to the Newton open polygon to find solutions y(x) to the nonlinear ordinary differential equation dy/dx = f(x, y)/g(x, y) near the point x = y = 0, where polynomials f and g vanish. A survey of other applications of the Newton (open) polygon was made by Chebotarev [1943]. The survey should be completed by mentioning the dual open polygons [Bruno 1979a] (see for example the Frommer open polygon [Frommer [1928]) and by attempts to reduce the solution of a system of algebraic equations of three or more coordinates to the plane Newton open polygon (see Section 10 of Chapter 2).

Sintsov [1898] to obtain expansions y(x), z(x) of the branches of solutions to an algebraic system of equations f1 (x, y, z) = f2...