asymptotic solution of an algebraic equation

ISSN 1064�5624, Doklady Mathematics, 2011, Vol. 84, No. 2, pp. 634–639. © Pleiades Publishing, Ltd., 2011.Original Russian Text © A.D. Bruno, A.B. Batkhin, 2011, published in Doklady Akademii Nauk, 2011, Vol. 440, No. 3, pp. 295–300.

634

We consider a polynomial in three variables near itssingular point, at which it vanishes together with itspartial derivatives. A method is proposed for comput�ing asymptotic expansions in terms of parameters forall the branches of the roots of this polynomial near itssingular point. There are three types of expansions.The computational method is based on spatial powergeometry. The method is applied to a sixth�degreepolynomial in three variables near its degenerate sin�gular point.

1. Let Q = (x, y, z) ∈ �3(�3), and let g(Q) be apolynomial. A point Q0 is called a singular point oforder k for the set � = {Q: g(Q) = 0} if all the partialderivatives of g up to the kth order vanish at this point.

Problem. For all the branches of the set �, findexpansions of the following three types in terms ofparameters near the singular point Q0 = 0:

Type 1: x = t s, y = t s, and z = t s ,

where bs, cs, and ds are constants.

Type 2: x = upv

q, y = upv

q, and z =

upv

q, where bpq, cpq, and dpq are constants and (p,

q) are integers lying in a sector of angle less than π.

Type 3: x = (u)vs, y = (u)vs, and z =

(u)vs, where βs(u), γs(u), and δs(u) are rational

functions of u and , and ψ(u) is a polynomial.

bs

s 1=

∞

∑ cs

s 1=

∞

∑ ds

s 1=

∞

∑

bpq∑ cpq∑dpq∑

βs

s 0=

∞

∑ γs

s 0=

∞

∑

δs

s 0=

∞

∑

ψ u( )

2. Let a finite sum (for example, a polynomial)

(1)

be given. Here, Q = (x, y, z) ∈ �3, R = (r1, r2, r3) ∈ �3,

QR = , and gR = const ∈ �. Each term insum (1) is associated with its vector power exponent R,and the entire sum (1) is associated with the set S of allvector power exponents of its terms, which is called thesupport of sum (1) or of the polynomial g(Q) and isdenoted by S(g). The convex hull of the support S(g) iscalled the Newton polyhedron of the sum g(Q) and isdenoted by Γ(g). The boundary ∂Γ of the polyhedron

Γ(g) consists of the generalized faces of differentdimensions d = 0, 1, 2. Here, j is the face index. Each

generalized face is associated with the truncatedsum

Let be the three�dimensional real space dual to�3, and let S = (s1, s2, s3) be points of this space. Given

points R ∈ �3 and S ∈ , the scalar product isdefined as

(2)

Specifically, the external normal Nk to the generalized

face is a point of . For it, the scalar product ⟨R,Nk⟩ on the support, i.e., over R ∈ S(g), reaches a max�

imum ck = ⟨R, Nk⟩ at points R ∈ ∩ S, i.e., at points

belonging to the generalized face . The set of all

those points S ∈ for which scalar product (2)reaches a maximum over all points R ∈ S(g) at points

R ∈ is called the normal cone of the generalized

face and is denoted by .

g Q( ) gRQRover R∑ S,∈=

xr1y

r2zr3

Γjd( )

Γjd( )

gjd( ) Q( ) gRQR

over R Γjd( ) S g( ).∩∈∑=

�*3

�*3

R S,⟨ ⟩ r1s1 r2s2 r3s3.+ +=

Γkd( )

�*3

Γkd( )

Γkd( )

�*3

Γkd( )

Γkd( ) Uk

d( )

Asymptotic Solution of an Algebraic EquationA. D. Bruno and A. B. Batkhin

Presented by Academician D.V. Anosov April 28, 2011

Received May 5, 2011

DOI: 10.1134/S1064562411060160

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russiae�mail: [email protected], [email protected], [email protected]

MATHEMATICS

DOKLADY MATHEMATICS Vol. 84 No. 2 2011

ASYMPTOTIC SOLUTION OF AN ALGEBRAIC EQUATION 635

Theorem 1. If, as t → ∞, the curve

(3)

where b, c, d, and si are constants, belongs to the set �

and the vector S = (s1, s2, s3) belongs to , then the

first approximation x = b , y = c , and z = d of

curve (3) satisfies the truncated equation (Q) = 0.

For the vertex , the truncated sum consistsof a single term. Such truncations are of no interestand are not addressed below. We consider only the

truncated sums corresponding to the edges

and the faces . The power transformations aredefined as

(4)

where lnQ = (lnx, lny, lnz)T, lnQ1 = (lnx1, lny1,lnz1)

T, and B is a nonsingular 3 × 3 matrix (bij) withrational elements bij (which are frequently integers).Here, ln means log. The matrix B with integer ele�ments such that detB = ±1 is called unimodular.Under power transformation (4), the monomial QR

turns into a monomial , where = BTRT. Powertransformations (4) and multiplications of a polyno�mial by a monomial (which corresponds to paralleltranslations of the support) induce an affine geometryin the space �3 of vector power exponents.

Theorem 2. For the face , there exists powertransformation (4) with a unimodular matrix B that

maps the truncated sum (Q) to a sum of d coordi�

nates; i.e., (Q) = h(Q1), where h(Q1) = h(x1) ifd = 1 and h(Q1) = h(x1, y1) if d = 2. Here, R = (r1, r2, r3).

The cone of the problem K is the set of vectors S =

(s1, s2, s3) ∈ such that curves of form (3) fill thepart of the space (x, y, z) to be studied. The problem inSection 1 is associated with the cone of the problem

K = {S = (s1, s2, s3): S < 0} located in , since x, y,z → 0. If x → ∞, then s1 > 0 in the cone of the prob�lem K. All these objects were introduced in [1,Chapters I, II].

3. Now we describe the steps of the procedure forsolving the problem.

Step 1. Compute the support S(g), the polyhedron

Γ(g), its two�dimensional faces , and their external

normals Nj. Compute the normal cones of the

edges using the normals Nj.

x bts1 1 o t( )+( ), y ct

s2 1 o t( )+( ),= =

z dts3 1 o t( )+( ),=

Ukd( )

ts1 t

s2 ts3

gkd( )

Γj0( ) gj

0( )

Γj1( )

Γj2( )

Qln B Q1,ln=

Q1

R1 R1T

Γjd( )

gjd( )

gjd( ) Q1

R

�*3

�*3

Γj2( )

Uk1( )

Γk1( )

Step 2. Select all those edges and faces whose normal cones intersect the cone of the problemK. For this purpose, it is sufficient to choose all the

faces for which the external normal Nj intersects

the cone of the problem K and to add all the edges of these faces.

Step 3. For each selected edge , perform thepower transformation from Theorem 2 to obtain a

truncated equation of the form h(x1) = 0. Let beone of its roots. Perform the power transformationQ → Q1 in the complete equation to obtain a polyno�mial g1(Q1) = g(Q).

If x1 is a simple root of the equation h(x1) = 0, then,by Theorem 1.1 from [1, Chapter II], it is associated

with an expansion of the form x1 – = ,

where apq are constants and (p, q) are integers lying insome sector [2, Chapter I, Section 3.4]. In the coordi�nates Q, it gives an expansion of type 2.

If is a multiple root of the equation h(x1) = 0,then construct the Newton polyhedron for the poly�nomial g1(Q1), compute a new cone of the problem K1

according to certain rules, and continue the computa�tions described.

Step 4. For each selected face , perform thepower transformation from Theorem 2 to obtain atruncated equation of the form h(x1, y1) = 0. Decom�

pose h(x1, y1) into irreducible factors. Let (x1, y1)be such a factor and ρ be the genus of the curve

(x1, y1) = 0.

If ρ = 0, then there exists a birational uniformiza�tion of this curve x1 = ξ(y2), y1 = η(y2). Make the sub�

stitutions x1 = ξ(y2) + x2 and y = η(y2). Then isdivided by x2. Pass to the variables x2, y2, and z2 = z1 inthe complete polynomial to obtain g2(Q2).

If (x1, y1) is a simple factor in h(x1, y1), then, byTheorem 1.1 from [1, Chapter II], in the completepolynomial g2(Q2) the solutions are expanded in aseries of the form

(5)

where αs(y2) are rational functions of y2. In the originalcoordinates, it is associated with an expansion of

type 3. If (x1, y1) is a multiple factor in h(x1, y1), thenconstruct the Newton polyhedron for g2(Q2), computethe cone of the problem K2, and continue the proce�dure.

Γk1( ) Γj

2( )

Γj2( )

Γk1( )

Γk1( )

x10

x10 apqy1

pz1q

∑

x10

Γj2( )

h

h

h

h

x2 αs y2( )z2s,

s 1=

∞

∑=

h

636


BRUNO, BATKHIN

If ρ = 1 (the case of an elliptic curve), then thereexists a birational substitution x1, y1 → x2, y2 that

reduces (x1, y1) to the form – ψ(y2), where ψ is apolynomial of order 3 or 4. If ρ > 1, then the cases of ahyperelliptic curve and a nonhyperelliptic curve haveto be distinguished. A hyperelliptic curve is biration�ally equivalent (x1, y1) → (x2, y2) to the normal form

– ψ(y2), where ψ is a polynomial of order 2ρ + 1 or

2ρ + 2 [3, Chapter III, Section 6]. If the factor inh is simple, then the solutions of the equation g2(Q2) = 0can be expanded in series (5), where αs is a rational

function of y2 and . In the original coordi�

nates, we obtain an expansion of type 3. If the factor in h is multiple, then continue the computations asdescribed above.

At each cycle consisting of Steps 1–4, the singular�ity simplifies and can be resolved after a finite numberof cycles.

Note some specific points of this procedure.

(a) If the truncated sum (Q1) is a power expo�nent of some sum ϕ(Q1), then, instead of Steps 3 and4, it is better to introduce the new variable x2 = ϕ(Q1)and not to change y1 and z1.

(b) For the asymptotic analysis of the set � as Q → ∞,perform Step 2, etc. It is the only case where all those

faces of the polyhedra Γ(g) are selected for whichthe external normals are positive: Nj > 0, since the coneof the problem is K = {S > 0}.

(c) The truncated sum corresponding to the

edge containing only two points of the support hasno multiple roots. The simple roots of the polynomial

corresponding to the edge do not need to beconsidered, since they are found in the families ofroots of the truncated polynomials corresponding to

the faces adjacent to the edge .

(d) The end of the procedure for computing expan�sions of singular points or multiple roots correspondsto an “open hole” in the polyhedron, which is associ�ated with a face parallel to the coordinate axis with anexternal normal vector of the form N = –(n, m, 0),where n, m ∈ N.

(e) In the case of an infinite expansion, the compu�tations can be terminated at a cycle where the face hasa fixed edge and the point opposite to this edge isalready far away (“stabilization”).

There are two cases in the procedure described.(i) If the truncated polynomial contains the linear

part in one of the variables, then an analogue of theimplicit function theorem is applicable and an expan�

h x22

x22

h

ψ y2( )

h

gjd( )

Γj2( )

gk1( )

Γk1( )

gk1( ) Γk

1( )

Γk1( )

sion of the set of roots of the entire polynomial can becomputed.

(ii) If the truncated polynomial does not containthe linear part with respect to any of the variables, thenpower transformation (4) is applied to the entire poly�nomial, its Newton polyhedron is calculated, and anew truncated polynomial is considered taking intoaccount the new cone of the problem K.

An asymptotic description of the subset of singularpoints of the set � is obtained by applying the sameprocedure, but singular points have to be chosen ineach truncated equation. As a result, we obtain anexpansion of type 1.

Theorem 3. If a procedure consisting of repeatedsequences of Steps 1–4 finds all the indicated faces andedges and all the roots of the corresponding truncatedequations at each step and all the available solutions ofpositive genus for the two�dimensional truncated equa�tions are elliptic or hyperelliptic and simple, then all thecomponents of the set � near the original point Q0 arelocally described in the form of expansions of types 1–3.

A similar procedure yields asymptotic expansionsof the branches of the set � near the point Q0 withcoordinates x0, y0, z0 = 0 or ∞ and near the coordinateaxes or coordinate planes consisting of singular points.Finally, the procedure described can be extended to analgebraic equation in n variables and to a system ofalgebraic equations (see [1, Chapters I, II]). For thecase of n = 2, see [2, Chapter I]. The procedure israther cumbersome and is better executed using acomputer algebra system after developing correspond�ing software codes. This has been done by the authors.

4. As an example, we consider expansions of thebranches of the set � for the polynomial g(Q) studiedin [4, 5] in the context of the stability analysis of agyroscopic problem. The global structure of the set �was analyzed in [4, Sections 4.4, 4.10]. Below, we onlyrecall the results.

The set � has four singular points of the secondorder:

Each of them is the intersection point of a pair of fam�ilies of first�order singular points. These families aretwo parabolas �1 (passes through the points Q0 and Q1)and �2 (passes through the points Q2 and Q3) and acurvilinear quadrilateral � passing through all the sin�gular points of the second order. The polynomial g(Q) is

Q0 0, Q1 2– 2 2, ,( ),= =

Q272�� 7

2�� 6, ,⎝ ⎠

⎛ ⎞ , Q352�� 3

2�� 2, ,–⎝ ⎠

⎛ ⎞ .= =

512z6 4352z5y– 768z5x– 14848z4y2 5376z4yx+ +

+ 512z4x225408z3y3– 14656z3y2x– 2752z3yx2–



(6)

The parametric representations of the families �1, 2

and � are

(7)

Simple points of the set � lie on a ruled surface thatis parametrically represented as

The set � was locally analyzed near the parabolas �1, 2

and the curve � in [4, Sections 4.6, 4.7] and at infinityand near the singular point Q0 in [6, Sections 2.2, 2.3].Below, we obtain expansions of types 1 and 3 for thebranches of the set � near the second�order singularpoint Q0.

– 192z3x321800z2y4

19168z2y3x+ +

+ 5360z2y2x2 736z2yx3 40z2x4 7500zy5–+ +

– 11700zy4x 4376zy3x2– 904zy2x3– 92zyx4–

– 4zx5 2500y5x 1200y4x2 344y3x3 48y2x4+ + + +

+ 4yx5 256z5– 2880z4y 1344z4x 14976z3y2–+ +

– 6720z3yx 1344z3x2– 37928z2y313816z2y2x+ +

+ 5144z2yx2 456z2x345120zy4– 14464zy3x–+

– 6784zy2x2 1152zyx3– 64zx4– 20250y5+

+ 6490y4x 3156y3x2 740y2x3 82yx4 2x5+ + + +

+ 1872z4 2016z3y 5088z3x– 35496z2y2–+

+ 15888z2yx 2200z2x267608zy3

12936zy2x–+ +

– 5176zyx2 344zx3– 37827y4– 828y3x 2782y2x2+ +

+ 412yx3 13x413824z3– 62208z2y 6912z2x+ + +

– 93312zy220736zyx– 1152zx2– 46656y3+

+ 15552y2x 1728yx2 64x3.+ +

�1: x 2t2, y– 4t– 2t2

, z– 6t– 4t2;–= = =

�2: x 3v v2

2��, y+ 3 v 1+( )2+

2��,= =

z 2 v 1+( )2; t v, �.∈+=

�:

x 2 ϕ 4 ϕcos2

1–( )sin4

3 2ϕsin3

–=

y 2 6 ϕcos4

8 ϕcos6

– 2ϕsin3

–+=

z 2 6 ϕcos2

8 ϕcos6

– 2 2ϕ,sin3

–+=⎩⎪⎨⎪⎧

0 ϕ π.<≤

�

x 3 u 1+( ) 2ϕsin– 2u ϕsin2 2ϕsin

2

2��,+ +=

y u 1+( ) 2ϕsin– 2u ϕcos2 2ϕsin

2

2�� 2,+ + +=

z 2 u 1+( ) 2ϕsin– u 1 2 ϕcos2

+( ) 2ϕsin2

3,+ ++=

0 ϕ π, v �.∈<≤

We define the cone of the problem K0 = {S < 0}, cal�culate the Newton polyhedron Γ(g) of polynomial (6)(see Fig. 1) and its external normals N0j, and chooseonly those faces whose normals belong to K0. The only

face of this kind is , and the corresponding trun�

cated polynomial is = 64(x + 9y – 6z)3.

Following remark (a) in Section 3, we make thesubstitutions x = 6z1 – 9y1 + x1, y = y1, and z = z1 to

obtain the polynomial g1(Q1) g(Q) and the newcone of the problem K1 = {S: s1 < s2 = s3 < 0}. Again,we compute the Newton polyhedron Γ(g1) and itsexternal normals N1j to the faces, among which there

is only one face with the normal vector N11 = (–4,–3, –3) ∈ K1, which corresponds to the truncated

polynomial = 64 – 432(3y1 – 2z1)4. The face

is a triangle, and two of its edges contain only twopoints of the support each. According to remark (c) inSection 3, these edges are not considered. The third

edge is associated with the factored truncated

polynomial = –432(3y1 – 2z1)4. This edge is adja�

cent to the face with the normal N16 = (–1, 0, 0).Therefore, we make the substitutions x1 = x2, y1 = y2,

and z1 = + z2 to obtain the polynomial g2(Q2)

g1(Q1). The new cone of the problem is K2 = {S =μ1N11 + μ2N16 + λ1(0, 0, – 1)}, where μ1, 2 ≥ 0, μ1 +μ2 > 0, and λ1 > 0. We compute the external normalsN2j to the faces of the new Newton polyhedron Γ(g2)

Γ012( )

g012( )

=def

Γ112( )

g112( ) x1

3 Γ112( )

Γ111( )

g111( )

Γ162( )

3y2

2�� =

def

5

4

3

2

1

6Pz

0 1 2 3 4 5

12453

Γ01(2)

PxPy

Fig. 1. Newton polyhedron for the polynomial g(Q). Thefaces with negative external normals are shown.

638


BRUNO, BATKHIN

(see Fig. 2). Among them, only two, N21 = –(4, 2, 3)and N22 = –(2, 0, 1), lie in the cone of the problem K2.

According to remark (d) in Section 3, the face is associated with a “hole” in the polyhedron, wherethe expansion procedure terminates. This face is asso�ciated with the straight line �1 = {x2 = z2 = 0}, which

belongs to the ruled surface . The face is associ�

ated with the polynomial = ⎯2 (y2 – 2) (x2y2 +

8 ). For small y2, it has the multiple root y2 = 0,which corresponds to the point Q0. The root of the last

factor of the polynomial corresponds to the part of

adjacent to the line �1.

The other chosen face is associated with thetruncation

(8)

This face is a quadrilateral, and three of its edgescontain only two points of the support each. Accord�ing to remark (c) in Section 3, they are not considered.

The fourth edge contains three points of the sup�port, and the corresponding truncated polynomial

= 4x2(4x2 – )2 contains a square multiple.Therefore, according to the general procedure, we

Γ222( )

� Γ222( )

g222( ) y2

3

z22

g222( )

�

Γ212( )

g212( ) 32z2

2y23 6912z2

4– 1152z22y2x2– 4x2 4x2 y2

2–( )2.+=

Γ212( )

Γ211( )

g211( ) y2

2

make the substitutions x2 = x3 + , y2 = y3, and z2 =

z3 to obtain the new polynomial g3(Q3) g2(Q2) andthe new cone of the problem K3 = {S = μ1N21 +μ2N25 + λ1(–1, 0, 0)}, where N25 = (0, 0, –1) is the

normal to the face adjacent to the edge . Forthe polynomial g3(Q3), we again compute the Newtonpolyhedron (see Fig. 3), its external normals N3j, etc.Such computations continue indefinitely, and the fol�lowing assertions hold true at each step: (i) there isonly one suitable face with a normal from the cone ofthe problem; (ii) this face is a triangle with a fixed edgeand a vertex lying on the axis; and (iii) according toremark (e) in Section 3, we have “stabilization” of theexpansion. Terminating the procedure at the fifth stepyields the following asymptotic expansion of the fam�

ily �1 near Q0: x = – 2t2 – 6t3 – – 3t5 – , y =

⎯4t – 2t2, z = –6t – 4t2 – t3 – (see (7)). This is an

expansion of type 1.

Let us show that, in the first approximation, trun�cated polynomial (8) gives an expansion of the ruled

surface near the point Q0. For this purpose, we per�form the power transformation

(9)

y32

4��

=def

Γ252( ) Γ21

1( )

13t4

2�� t6

2��

t4

4��

�

x2 ξ 1– η 2– ζ4, y2 ξ 1– η 1– ζ2

, z2 ξ 1– η 1– ζ3,= = =

5

4

3

2

1

6

Pz2

0 1 2 3 4 5

1245

3

Γ21(1)

Py2Px2

Γ21(2)

Γ22(2)

5

4

3

2

1

6

Pz3

0 12

34 5

5Py3

Px3

Γ13(2)

10

Fig. 2. Newton polyhedron for the polynomial g2(Q2). Thefaces with negative external normals are shown.

Fig. 3. Newton polyhedron for the polynomial g3(Q3).



which is defined by the unimodular matrix B =

. After factorization, polynomial (8) is

reduced to a polynomial in two variables: (ξ, η) =1728ξη2 – 16ξ2 + 288ξη + 8ξ – 8η – 1. Its rootdefines a plane curve of genus 0 (see Fig. 4), which canbe birationally parametrized as ξ = Ξ(w), η = Η(w).This parametrization defines the substitution

(10)

as described in Section 3 (Step 4) for the case ρ = 0. Tofind the first term of expansion (5), substitutions (9)and (10) are sequentially made in the complete poly�nomial g2(Q2) and then a common multiple is found,while the remaining polynomial is represented as onein two variables ξ1 and ζ1 with polynomial coefficientsin w. Using ξ1 and η1, we calculate the Newton poly�gon for this polynomial and find an edge with a nega�tive external normal. Then the initial term in expan�sion (5) is determined by solving the truncated poly�nomial corresponding to the chosen edge of theNewton polygon. In the case under study, this term is

α1(w) = – . Then ξ1 = α1(w)ζ1 +

o(ζ1). Sequentially combining the substitutions, weobtain the initial terms of the expansion of type 3

for the ruled surface at the point Q0: x = 12Ω(w) ,

y = 2Ω(w) , and z = 3Ω(w) , where Ω =

.

The singular point (–1/12, –1/9) of the curve

(ξ, η) = 0 corresponds to a one�parameter family� of first�order singular points, whose expansion nearthe point Q0 was obtained in [6].

Theorem 4. The following objects of the set arefound near the second�order singular point Q0:

(i) the straight line �1, which is the generatrix line of

the ruled surface ;

1– 2– 4

1– 1– 2

1– 1– 3⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

h212( )

ξ ξ128w 3+( ) 12w 1+( )

64 9w 1+( )2��+ ,=

η 8w 1+( )2

16 9w 1+( ) 12w 1+( )��, ζ ζ1,= =

10w 1+( ) 12w 1+( )2 8w 1+( ) 9w 1+( )��

� ζ13

ζ12 ζ1

2

512 9w 1+( )3

28w 3+( ) 8w 1+( )2��

h212( )

�

�

(ii) type 1 expansions of the families �1 and � offirst�order singular points;

(iii) a type 3 expansion of the ruled surface .

ACKNOWLEDGMENTS

This work was supported by the Russian Founda�tion for Basic Research, project no. 11�01�00023.

REFERENCES1. A. D. Bruno, Power Geometry in Algebraic and Differen�

tial Equations (Fizmatlit, Moscow, 1998; Elsevier Sci�ence, Amsterdam, 2000).

2. A. D. Bruno, Local Methods in Nonlinear DifferentialEquations (Nauka, Moscow, 1979; Springer�Verlag,Berlin, 1989).

3. I. R. Shafarevich, Basic Algebraic Geometry (Springer�Verlag, Berlin, 1974; MTsNMO, Moscow, 2007).

4. A. D. Bruno, A. B. Batkhin, and V. P. Varin, PreprintNo. 4, IPM RAN (Keldysh Inst. of Applied Mathemat�ics, Russian Academy of Sciences, Moscow, 2010).

5. A. D. Bruno, A. B. Batkhin, and V. P. Varin, PreprintNo. 23, IPM RAN (Keldysh Inst. of Applied Mathe�matics, Russian Academy of Sciences, Moscow, 2010).

6. A. D. Bruno and A. B. Batkhin, Preprint No. 10, IPMRAN (Keldysh Inst. of Applied Mathematics, RussianAcademy of Sciences, Moscow, 2011).

�

0.1

0

−0.1

η

1 2 ξ

−0.2

−0.3

Fig. 4. Curve (ξ, η) = 0 obtained using power transfor�mation (9).

h212( )

asymptotic solution of an algebraic equation

Documents