POWER GEOMETRY IN LOCAL RESOLUTION OF SINGULARITIES OF AN ALGEBRAIC CURVE Текст научной статьи по специальности «Математика»

PHYSICS AND MATHEMATICS

POWER GEOMETRY IN LOCAL RESOLUTION OF SINGULARITIES OF AN ALGEBRAIC CURVE

Akhmadjon Soleev

Doctor of science, professor Samarkand State University, Uzbekistan, ORCID ID: https://orcid.org/0000-0003-4921-2349

DOI: https://doi.org/10.31435/rsglobal_ws/30062020/7100

ARTICLE INFO

Received: 25 April 2020 Accepted: 10 June 2020 Published: 30 June 2020

KEYWORDS

power geometry, space curve, singular point, Newton polyhedron, normal cone, power transformations. MSC (2010): 14B05, 53A04, 58E07, 58K45, 58K55.

ABSTRACT

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.

Citation: Akhmadjon Soleev. (2020) Power Geometry in Local Resolution of Singularities of an Algebraic Curve. World Science. 6(58), Vol.1. doi: 10.31435/rsglobal_ws/30062020/7100

Copyright: © 2020 Akhmadjon Soleev. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

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 f(d) = 0 by means of generalized faces of the convex polyhedron r (f), which is a generalization of the Newton polyhedron. The first term of the expansion of a solution to the initial equation f = 0 is a solution to the corresponding truncated equation /^(d) = 0.

2. Finding solutions to a truncated equation = 0, which is quasi homogenous. Using power and logarithmic transformations of coordinates we can reduce the equation f(d) = 0 to such a

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.

Algebraic equations [2, 3].

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 computation of asymptotic expansions of all branches of the set of roots of the polynomial near the mentioned singular point. Now there are three types of expansions. The method of computation Is based on space Power Geometry. All examples are for polynomials in two variables.

Let X = (x1, x2, x3) c R3or C3 and f(X) be a polynomial. X0 is called to be singular for the set T = [X • f(X) = 0} if all the partial derivatives of the first order of the polynomial f vanish at the point X0 and f(X0) = 0.

Consider the following problem. Near the singular point X0 for each branch of the set T, find a parameter expansion of one of the following three types [6].

Type 1

Xi = ^ bkvk, x2 = ^ ckvk, x3 = ^ dkVk k=1 k=1 k=1

where bk, ck , dk are constants. Type 2

x1 = ^ bpqupvq, x2 = ^ cpqupvq , x3 = ^ dpqupvq ,

where bpq, cpq, dpq are constants and integer points (p,q) are in a sector with the angle less than n. Type 3

k=0 k=0 _ k=0

where fik(u), yk(u), Sk(u) are rational functions of u and ^\p(u), and is a polynomial in u. Objects and algorithms of Power Geometry. Let a finite sum be given (for example, a polynomial)

f(X)=£fQXQ over Q e S, (4.1)

where X = (x1, x2, x3) e R3, Q = (qt, q2, q3) e R3 and XQ = x^1x^2x33, fq = const e R .

To each of the summand off 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 r(f).

The boundary 3r of the polyhedron rf) consists of generalized faces of various dimensions d = 0, 1, 2. Here j is the number of a face. To each generalized face r(d^, we assign the truncated sum f(d)(X) = £ fQXQ over Q e r(d) n S(f).

Example 1. We consider the polynomial f(x,y) = x^ — xy2. Support Sf) consists from points Q\=(5,0), Q2=(0,5), Q3=(1,2).

The Newton polygon rf) is the triangle Q1 Q2 Q3 (figure 1). Edges and corresponding truncated polynomials are

r(x) • f^ = x5 — xy2, T« • f2(1) =y5 — xy2, T« • f3(1) = x5+y5 , Let R3 be a space dual to space R3 and S = (si, S2, s3) be points of this dual space. The scalar

product

CO

DO

DO

<0,5> = 9isi + 92*2 + 93*3 (4.2)

is defined for the points Q e R3 and S e Ef. Specifically, the normal external Nk to the generalized face r(d) is a point in Ef.

The scalar product <Q, Nfc> reaches the maximum value at the points Q e rjj,d) n S , i.e., at the points of the generalized face rjj,d). Moreover, set of all points 5e Ef, at which the scalar product (4.2) reaches the maximum over Q e S(/) exactly at points Q e r(d) , is called the normal cone of the generalized face r(d) and is denoted by U(d).

Example 2: (cont. of Example 1). For faces rjj,d) of the Newton polygon rf) of Figure 1, normal cones are shown in Figure 2.

For edge r(1) j = 1,2,3 normal cone t/(1) is a ray orthogonal to its edge. For vertex rj0) = Qy =

(i)

Ry, j = 1,2,3 normal cone is open sector between rays orthogonal to edges r- adjacent to vertex Rj. Theorem 1. If for t ^ to the curve

x1 = ¿tsi(l + (0)), x2 = cts2(i + (0)), x3 = dts3(i + (0)) (4.3)

where b, c, d and si are constants, belongs to the set Q, and the vector S = (s1, s2, s3)e U(d), then the first approximation x1 = btSl, x2 = cts2, x3 = dts3 of the curve (4.3) satisfies the truncated equation f(d

fkd)(X) = 0.

See the proof of the theorem in the paper [2, 3].

Fig. 1 Fig. 2

The truncated sum /^(0) corresponding to the vertex rj0) is a monomial. Such truncations are

(1)

of no interest and will not be considered. We will consider truncated sums corresponding to edges r ■

and faces rj2) only.

Power transformations have the form

logX = alogF, (4.4)

where logX = (logx1,logx2,logx3)T, logF = (logy1,logy2,logy3)T, a is a nondegenerate square 3 * 3 matrix (a;y) with rational elements a;y (they are often integer).

The monomial is transformed to the monomial by the power transformation (4.4), where = aTQT. Power transformations and multiplications of a polynomial by monomial generate the affine geometry in space R3 of vector power exponents of polynomial monomials. The matrix a with integer elements and det a = ±1 is called unimodular.

Theorem 2. For the face Tyd), there exists a power transformation (4.4) with a unimodular matrix a which transforms the truncated sum fj~d\X) into the sum in d coordinates, i.e. /^-(d)(X) = where h(F) = ^(y1) if d = 0, h(F) = ^(y2y3) if d = 2. Here Q' = ql, q2, 93e R3 and other coordinates y2,y3 for d = 1, for d = 2 are small. For the polynomial (X) the sum h(Y) is also polynomial. The proof of this theorem is similar to the proof of theorem 3 in the paper [2]. The cone of the problem K is a set of such vectors S = (s1, s2, s3)eR3 that curves of the form (4.3) fill the part of the space (x1, x2, x3), which must be studied. So, our initial problem corresponds

3. If y° is a simple root of the equation h(y1) = 0 then, according to Implicit Function

to the cone of the problem K = [S = (s1, s2, s3):S < 0} in R3, because x1, x2, x3 ^ 0. If x1 ^ then s1 > 0 in the cone of the problem K.

Example 3. For variables x, y near origin x = y = 0 cone of the problem is the quadrant III:

(2)

[K3 = s1s2 > 0}. In Figure 2 some cones of the problem Ki intersects several normal cones Uj . E.g.

K3 intersects U^, U(2) and U(°\ U(2\ U^. Ki intersects U^, Ui2). Let's give a step-by-step algorithm for solving the problem.

1. We compute the support Sf), the Newton polyhedron T(f), its two-dimensional faces T(2)and their external normal Nj. Using normal Nj we compute the normal cones Uj1 to edges T^.

2. We select all the edges T(1) and faces r(2\ which normal cones intersect the cone of the

(2)

problem K. It is enough to select all the faces T- , which external normal Nj intersect the cone of the

(1)

problem K, and then add all the edges Tk of these faces

(1)

a) For each of the selected edge Tk , we fulfill a power transformation X ^ Y of Theorem 2 and we get the truncated equation in a form h(y1) = 0.

b) We find its roots. Let y° be one of its roots.

c) We fulfill the power transformation X ^ Y in the whole polynomial f (X) and we get the polynomial f1 (F).

d) We make the shift z1 = y1 — y°, z2 = y2, z3 = y3 in the polynomial f1(Y) and get the polynomial f2 (Z).

)

Theorem, it corresponds to an expansion of the form y2 = Y^pqy^y^ where apq are constants. It

gives an expansion of type 2 in coordinates Y.

(1)

4. For each of the selected face Tk , we fulfill a power transformation X ^ Y of Theorem 2 and we get a truncated equation in the form h(y1,y2) = 0. We factorize h(y1,y2) = 0 into prime factors. Let h(y1,y2) = 0 be one of such factors and its algebraic curve has genus p.

5. If p = 0 then there exists birational uniformization y1 = ((z2), y2 = tf(z2) of this curve. We change variables y1 = ((z2) + z1, y2 = tj(z2) and then h is divided by z\. We change variables in the whole polynomial f (X) and get the polynomial f2 (Z) = f1 (Y) = f (X)

If h(y1,y2) is a simple factor of h(y1,y2) then roots of the polynomial f2(Z) are expanded into series of the form

Z1 = Y^=1ak(z2)z3^ (4.5)

where ak(z2) are rational functions of z2. It gives an expansion of type 3 in original coordinatesX.

If h(y1, y2) is a multiple factor of h(y1, y2) then we compute the Newton polyhedron of the polynomial f2(Z), compute the cone of problem K2 = [S: s2,s3 < 0} and continue computations.

6. If p =1 (elliptic curve), there exists the birational change of variables y1,y2 ^ z1,z2, transforming fi(y1, y2) = 0 into the normal form z2 — ty(y2), where y is a polynomial of order 3 or 4.

If p > 1, we distinguish hyper-elliptic and non hyper-elliptic curves. The hyper-elliptic curve is birationally equivalent y1,y2 ^ zx,z2 to its normal form z2 — ty(y2) , where y is a polynomial of order 2p + 1 or 2p + 2.

If factor h of h is simple we get expansions of solutions of equation f2(Z) = 0 into series (4.5), where ak are rational functions of z2 and We get the expansion of type 3 in original

coordinates X.

If fi(y1,y2) is a multiple factor of h(y1,y2) then we continue computation for f2(Z) as above. 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.

Example 4 (cont. of Examples 1-3).

1. For edge r(_ , we get a truncated equation x5 — xy2 = 0 î. e. y = ±x2. It is case 1, and this asymptotic form is continued into power expansion of branch

œ

y = ±x2 + ^ ôfcx2fc fc=2

near the origin x = y = 0 (figure 3).

2. For edge r21), we get a truncated equation y5 — xy2 = 0 î. e. y = ±x1/3. It is case 1, and these asymptotic forms are continued into power expansion branches

k

near the origin x = y = 0 (figure 4).

Fig. 3 (1)

y = ±Vx + ^ Ôfcx3 fc=2

Fig. 4

3. For edge r3 , we get truncated equation x3+y3 = 0. It has the simple factor x + y = 0, i.e y = — x. It is case 1, and the power expansion at infinity is

y = -x + ^ ôfcx-

k=2

*s + y5 - xy* = 0

CO

CO

k

Fig. 5

Figure 5 shows a general view of the equation /(x,y) = x5 + y5 — xy2. In the neighborhood of a singular point.

Asymptotic description of a subset of singular points of Q 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 Q adjacent to the starting point X0, in the form of expansions of types 1-3.

REFERENCES

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