POWER GEOMETRY IN LOCAL RESOLUTION OF SINGULARITIES OF AN ALGEBRAIC CURVE

The main goal of this work is to provide a consistent set of generalpurpose algorithms for analyzing singularities applicable to all types of equations. We present the main ideas and algorithms of power geometry and give an overview of some of its applications. We also present a procedure that allows us to distinguish all branches of a spatial curve near a singular point and calculate the parametric appearance of these branches with any degree of accuracy. For a specific case, we show how this algorithm works. KEYWORDS

Introduction. Many problems in mathematics, physics, biology, economics, and other sciences are reduced to nonlinear equations or 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 of 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 the analysis of solutions neighboring the singular one. Although different methods of such analysis were suggested for some special problems.
Main Part. We develop a new calculus based on Power Geometry [1,2,3,4]. Here we will consider only to compute local and asymptotic expansions of solutions to nonlinear equations of algebraic classes as well as to systems of such equations. But it can also be extended to other classes of nonlinear equations for such as differential, functional, integral, and integro-differential [7].
Ideas and algorithms are common for all classes of equations. Computation of asymptotic expansions of solutions consists of 3 following steps (we describe them for one equation f = 0).
1. Isolation of truncated equations ̂( ) = 0 by means of generalized faces of the convex polyhedron Г( ), which is a generalization of the Newton polyhedron. The first term of the expansion of a solution to the initial equation = 0 is a solution to the corresponding truncated equation ̂( ) = 0. simple form that can be solved. Among the solutions found we must select appropriate ones that give the first terms of asymptotic expansions. 3. Computation of the tail of the asymptotic expansion. Each term in the expansion is a solution of a linear equation that can be written down and solved.
Elements of plane Power Geometry were proposed by Newton for algebraic equations (1670). Space Power Geometry for a nonlinear autonomous system of ODEs was proposed by Bruno (1962) [1]. Thus, now it is exactly 50 years for the Newton polyhedron.
It is clear that this calculus cannot be mastered during this paper. We will try to summarize our ideas and in the next paper, we will consider this problem and give algorithms for nonlinear systems of algebraic equations.
In this paper, we consider a polynomial depending on three variables near its singular point where the polynomial vanishes with all its first partial derivatives. We propose a method of Consider the following problem. Near the singular point 0 for each branch of the set ℱ, find a parameter expansion of one of the following three types [6].
Type 1 where , , are constants. Type 2 where , с , are constants and integer points (p,q) are in a sector with the angle less than π. Type 3 where ( ), ( ), ( ) are rational functions of u and √ ( ), and ( ) is a polynomial in u.

Objects and algorithms of Power Geometry.
Let a finite sum be given (for example, a polynomial) To each of the summand of f the sum (4.1), we assign it vector power exponent Q, and to the whole sum (4.1), we assign the set of all vector power exponents of its terms, which is called the support of the sum (4.1) or of the polynomial f(X), and it is denoted by S(f). The convex hull of the support S(f) is called the Newton polyhedron of the sum f(X) and it is denoted by Γ(f).
The boundary ∂Γ of the polyhedron Γ(f) consists of generalized faces Г ( ) of various Here j is the number of a face. To each generalized face Г ( ) , we assign the truncated sum ̂( ) ( ) = ∑ Г ( ) ∩ ( ).
The Newton polygon Γ(f) is the triangle Q1 Q2 Q3 (figure 1). Edges and corresponding truncated polynomials are Let ℝ * 3 be a space dual to space ℝ 3 and = ( 1 , 2 , 3 ) be points of this dual space. The scalar product 〈 , 〉 = 1 1 + 2 2 + 3 3 (4.2) is defined for the points ℝ 3 and ℝ * 3 . Specifically, the normal external Nk to the generalized The scalar product 〈 , 〉 reaches the maximum value at the points Г ( ) ∩ , i.e., at the points of the generalized face Г ( ) . Moreover, set of all points ℝ * 3 , at which the scalar product See the proof of the theorem in the paper [2,3]. The truncated sum ̂( 0) corresponding to the vertex Г (0) is a monomial. Such truncations are of no interest and will not be considered. We will consider truncated sums corresponding to edges Г (1) and faces Г (2) only.

1.
We compute the support S(f), the Newton polyhedron Γ(f), its two-dimensional faces Г (2) and their external normal Nj. Using normal Nj we compute the normal cones 1 to edges Г (1) .

2.
We select all the edges Г (1) and faces Г (2) , which normal cones intersect the cone of the problem K. It is enough to select all the faces Г (2) , which external normal Nj intersect the cone of the problem K, and then add all the edges Г (1) of these faces a) For each of the selected edge Г (1) , we fulfill a power transformation X → Y of Theorem 2 and we get the truncated equation in a form ℎ( 1 ) = 0. b) We find its roots. Let 1 0 be one of its roots. c) We fulfill the power transformation X → Y in the whole polynomial f (X) and we get the polynomial 1 ( ). d) We make the shift 1 = 1 − 1 0 , 2 = 2 , 3 = 3 in the polynomial 1 ( ) and get the polynomial 2 ( ). (4.5) where ( 2 ) are rational functions of 2 . It gives an expansion of type 3 in original coordinates X.
If factor h of h is simple we get expansions of solutions of equation 2 ( ) = 0 into series (4.5), where are rational functions of 2 and √ ( 2 ). We get the expansion of type 3 in original coordinates X.
In this procedure, we distinguish two cases: 1. Truncated polynomial contains a linear part of one of the variables. The generalization of the Implicit Function Theorem is applicable and it is possible to compute parametric expansion of a set of roots of a full polynomial.
2. Truncated polynomial does not contain a linear part of any variable. Then the Newton polyhedron for a full polynomial must be built and we must consider new truncated polynomials taking into account the new cone of the problem K.   Asymptotic description of a subset of singular points of Ω can be obtained by the same procedure, but we have to select only singular points in each truncated equation. As a result, we obtain expansions of type one.
So we got the following result: If we perform calculations for 1-4 using this procedure, then at each step we find all the roots of the corresponding truncated equations, and find all the curves of the roots of the truncated equations with a positive native elliptic or hyperelliptic, we get a local description of each component of the set Ω adjacent to the starting point 0 , in the form of expansions of types 1-3.