- Email: [email protected]
Computation and applications of the Newton polyhedrons
Mathematics and Computers in Simulation 57 (2001) 155–160
Computation and applications of the Newton polyhedrons Alexander B. Aranson Department of Mathematics, Keldysh Institute of Applied Mathematics, Miusskaja Sq. 4, Moscow 125047, Russia
Abstract We consider the multivariate Laurent polynomial f (X) = aQ X Q , Q ∈ D with coefficients aQ ∈ R or C and D is some set in Zn . The set D = D(f ) = {Q: aQ = 0} is called the support of the polynomial f (X). The convex hull M = M(f ) of the set D is called the Newton polyhedron of the polynomial f (X). There are important correspondences between properties of the polynomial f (X) and of its Newton polyhedron M(f ) that were studied by Bruno, Soleev, Khovansgfkii and others. We propose algorithms and the computer program for computation of the Newton polyhedron of any polynomial, and for computation of all elements of this polyhedron (vertices, edges, faces, etc.). © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Systems of differential polynomials; Newton polyhedron; Intersections of normal cones; Truncated systems
We consider the Laurent polynomial f (X) = aQ XQ , Q ∈ D
(1) q
q
q
where X = (x1 , x2 , . . . , xn ) ∈ Rn or Cn , Q = (q1 , q2 , . . . , qn ) ∈ Zn , monomial X Q = x1 1 x2 2 . . . xnn , coefficients aQ ∈ R or C and D is some set in Zn . The set D = D(f ) = {Q: aQ = 0} is called the support of the polynomial f (X). The convex hull M = M(f ) of the set D is called the Newton polyhedron of the polynomial f (X). The boundary ∂M of the polyhedron M consists of faces Γk(d) of different dimension d(0 ≤ d < n). Zero dimensional faces Γk(0) are vertices of the polyhedron M, one dimensional faces are its edges etc. Let Rn∗ denote the dual space such that for P = (p1 , . . . , pn ) ∈ Rn∗ and Q = (q1 , . . . , qn ) ∈ Rn with the scalar product P , Q = p1 q1 + . . . + pn qn . For a fixed vector P = 0, let DP denote such subset of the set D that in each point Q ∈ DP the scalar product P , Q has the maximal value cP among all Q ∈ D, that is P , Qj = cP
for Qj ∈ DP
and P , Qj < cP
for Qj ∈ D \ DP .
E-mail address: [email protected] (A.B. Aranson). 0378-4754/01/$ – see front matter © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 7 5 4 ( 0 1 ) 0 0 3 3 5 - 4
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According to (2), to each vector P ∈ Rn∗ there corresponds a supporting to the set D hyperplane LP = {Q: P , Q = cP } lying in Rn . It intersects the polyhedron M along a face Γk(d) , where d is the dimension of the face and k is its number. The set Uk(d) of all vectors P ∈ Rn∗ such, that LP ∩ M = Γk(d) , is called the normal cone of the face Γk(d) . Let dim M = n. The normal cone Uk(n−1) of the hyperface Γk(n−1) is a ray orthogonal to the face and directed out of the polyhedron M. The normal cone Uk(n−2) is a sector bounded by rays Ul(n−1) and Um(n−1) if Γk(n−2) = Γl(n−1) ∩ Γm(n−1) etc. The union U of normal cones of all faces is Rn∗ \ {0}. Let Dk(d) = Γk(d) ∩ D be a boundary subset of the set D. Lemma 1. [1,2]. If P ∈ Uk(d) , then DP = Dk(d) . The sum fj XQj for Qj ∈ Dk(d) fˆk(d) (X) =
(3)
is called the truncation of the polynomial (1). Now we consider a curve of the form xi = bi τ pi (1 + o(1)),
bi = 0,
i = 1, . . . , n,
(4)
where τ → ∞. On such curve a monomial XQ = B Q τ P ,Q (1 + o(1)), where B = (b1 , . . . , bn ) and P = (p1 , . . . , pn ). If P ∈ Uk(d) , then on the curve (4), the polynomial (1) is f (X) = fˆk(d) (B)τ cP +τ cP o(1), where cP is from (2). Thus, the truncation (3) is the first approximation of the polynomial f (X) on curves (4) with P ∈ Uk(d) . If curve (4) is a solution of the equation f (X) = 0 then fˆk(d) (B) = 0. That is we have Theorem 1. [1,2]. Let the curve (4) be a solution of the equation f (X) = 0 and P ∈ Uk(d) . Then the first approximation xi = bi τ pi , i = 1, . . . , n of the solution (4) is a solution of the corresponding truncated equation fˆk(d) (X) = 0. This approach allows to find solutions of the equation f = 0 with any accuracy [1]. For other applications of the Newton polyhedron in various mathematical fields see in [1]. Example 1. Let n = 2 and f = x13 +x23 −2x1 x2 , then the set D = {Q1 , Q2 , Q3 } = {(3, 0), (0, 3), (1, 1)}. Its boundary subset are D1(0) = Q1 ,
D2(0) = Q3 ,
D1(1) = {Q1 , Q3 },
D3(0) = Q2
D2(1) = {Q2 , Q3 },
D3(1) = {Q1 , Q2 }
and some of their normal cones are U1(1) = −λ(1, 2),
U2(1) = −λ(2, 1),
U3(1) = −λ(1, 1)
(λ > 0).
The Newton polyhedron and normal cones see in Fig 1. Among truncations fˆk(d) there are fˆ1(0) = x13 ,
fˆ2(0) = −2x1 x2 ,
fˆ3(0) = x23 ;
fˆ1(1) = x13 − 2x1 x2 ,
fˆ2(1) = x23 − 2x1 x2 .
A.B. Aranson / Mathematics and Computers in Simulation 57 (2001) 155–160
157
Fig. 1. The Newton polyhedron (a) and normal cones of its faces (b) for Example 1.
Roots of the truncated equation fˆ1(1) = 0 are x2 = 1/2x12 , i.e. the first approximation of a branch of the f = 0 near X = (x1 , x2 ) = 0. The roots of the equation fˆ2(1) = 0 give the first approximation of another branch of the curve near the origin. The aim of the talk is to describe the algorithm and computer program for the computation of all boundary subsets Dk(d) for any given finite set D = {Q1 , . . . , Qm } ⊂ Zn and their normal cones Uk(d) . The system of equations and inequalities (2) determines two objects: the boundary subset DP and its normal cone UP . Thus, it is necessary to find solution of every such system (2) to select all DP and UP . But all these systems can be presented as the single system of linear weak inequalities P , Qj ≤ cP ,
j = 1, . . . , m.
(5)
˜ = (Q, qn+1 ) where additional Instead of vectors P and Q we consider vectors P˜ = (P , pn+1 ) and Q coordinate pn+1 = −cP and qn+1 = 1 [3]. Now the system (5) becomes the system of linear homogeneous weak inequalities ˜ j ≤ 0, P˜ , Q
j = 1, . . . , m.
(6)
The set of solutions to system (6) is a polyhedral convex cone U˜ ⊂ Rn+1 ∗ . The skeleton of the cone ˜ U˜ consists of some vectors P˜1 , . . . , P˜l ∈ Rn+1 , so the cone U is their conic hull. Each vector P˜k = ∗ (n−1) of the polyhedron M. Moreover to each (Pk , pk,n+1 ) where Pk is a normal vector to a hyperface Γk (d+1) ˜ ˜ of the cone U , there corresponds the normal cone Uk(d) of the face Γk(d) of the polyhedron face Uk M. In more details, let P˜k1 , . . . , P˜ks form the skeleton of the case U˜ k(d+1) ⊂ U˜ , then the boundary subset ˜ j = 0, i = 1, . . . , s}. Dk(d) = {Qj ∈ D: P˜ki , Q ˜ The vectors Pk of the skeleton of the cone U˜ can be obtained by Motzkin–Burger algorithm [1,4–6]. The algorithm consists of several steps. By the first step one obtains solutions of the first inequality, by the second step one take in account the second inequality etc. The geometry of this algorithm is the projection of skeleton vectors of the solutions of previous inequalities into the boundary of the half-space determined by the current inequality. ˜ 1 = (3, 0, −1), Q ˜ 2 = (0, 3, −1), Q ˜ 3 = (1, 1, −1). Example 2. (continuation of Example 1). Here Q Three steps of the Motzkin–Burger algorithm for the set D of Example 1 are shown in Figs. 2–4.
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A.B. Aranson / Mathematics and Computers in Simulation 57 (2001) 155–160
˜ 1 ≤ 0. The vector Q ˜ 1 ∈ R3 , the vector P˜11 ∈ R3∗ and Fig. 2. Step 1 for Example 2. The set of solutions to the inequality P˜ , Q ˜ 1 , are shown. part of the hyperplane H in R3∗ , with a basis B˜ 11 , B˜ 21 , which is normal to the vector Q
˜ 1 ≤ 0. The set of its solutions consists of a half-space, Step 1: We are solving the inequality P˜ , Q 1 ˜ which is bounded by plane H with basis B1 = (0, 1, 0) and B˜ 21 = (1, 0, −3). The vector P˜11 = (−1, 0, 0) belongs to the interior of the half-space (see Fig. 2). Step 2: We are calculating the intersection of the half-space, found in the step 1, with the half-space, ˜ 2 ≤ 0. This intersection is a dihedral angle with the vertex in the form of defined by the inequality P˜ , Q the line L. The basis of this line is the vector B˜ 12 = (−1, −1, 3). The vector P˜12 = (1, −10, −3) belongs to the interior of one face of the dihedral angle and the vector P˜22 = (−10, 1, −3) belongs to another face (see Fig. 3). Step 3: We are calculating the intersection of the dihedral angle, found in the step 2, with the half-space, ˜ 3 ≤ 0. Vectors P˜1 = (−2, −1, 3), P˜2 = (−1, −2, 3), P˜3 = (1, 1, −3) defined by the inequality P˜ , Q
˜ 1 ≤ 0, P˜ , Q ˜ 2 ≤ 0. Vectors Q ˜ 1, Q ˜ 2 ∈ R3 , Fig. 3. Step 2 for Example 2. The set of solutions to the system of inequalities P˜ , Q ˜ 1, Q ˜ 2 , and dihedral angle of solutions are shown. vectors P˜12 , P˜22 ∈ R3∗ , the line L with a basis B˜ 12 , which is normal to vectors Q
A.B. Aranson / Mathematics and Computers in Simulation 57 (2001) 155–160
159
˜ 1 ≤ 0, P˜ , Q ˜ 2 ≤ 0, P˜ , Q ˜ 3 ≤ 0. Vectors Fig. 4. Step 3 for Example 2. The set of solutions to the system of inequalities P˜ , Q ˜ 1, Q ˜ 2, Q ˜ 3 ∈ R3 and vectors P˜1 , P˜2 , P˜3 ∈ R3∗ which form the skeleton of the cone of solutions, are shown. Q
˜ 1 ≤ 0, P˜ , Q ˜ 2 ≤ 0, form the skeleton of the cone, which is the set of solutions of the system P˜ , Q ˜ 3 ≤ 0 (see Fig. 4). P˜ , Q I have realized the described algorithms as a computer program in the C++ language [3,4]. This program reads file with coordinates of all points of the set D and calculate following items: • All normal vectors Pk of hyperfaces Γk(n−1) of the polyhedron M and constants cP . • The corresponding matrix where the columns of this matrix correspond to the vectors Qj of the set D and rows correspond to normal vectors Pk of hyperfaces Γk(n−1) of M. In the intersection of the column with number j and the row with number k, we write the sign “+” if Qj ∈ Dk(n−1) and the sign “−” in the contrary case (see Table 1). • The list of all boundary subsets Dk(d) . For each of them, the program writes its dimension d, numbers j of all points Qj ∈ Dk(d) and numbers l of normal vectors Pl forming the skeleton of the normal cone Uk(d) (see Table 2). The program was used in a study of surface waves on water (with n = 6) ([1], Ch. V; and [7,8]) and in a study of the discrete models of the Boltzman equation (with n = 25) [9]. The algorithm and the Table 1 The corresponding table for Example 1
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Table 2 The list of all boundary subsets Dk(d) for Example 1 k
d
j
l
1 2 3 1 2 3
1 1 1 0 0 0
1,3 2,3 1,2 1 2 3
1 2 3 1,3 2,3 1,2
computer program described above can be applied to other fields of mathematics, in particularly, for a selection of Pareto-optimal sets in the multicriterial optimization [10]. Acknowledgements The author thanks Prof. A.D. Bruno for his supervision of the work. The work was supported by RFBR, Grant 99-01-01063. References [1] A.D. Bruno, Power Geometry in Algebraic and Differential Equations, Fizmatgiz, Moscow, 1998, p. 288 (in Russian, its English translation was published by Elsevier, 2000). [2] A.D. Bruno, Power geometry, J. Dynamic. Contr. Syst. 3 (4) (1997) 471–491. [3] A.B. Aranson, Computation of the Newton polyhedron, in: Proceedings of the International Conference Devoted to 175th Anniversary of P.L. Chebyshev, Moscow State University, 1 (1996) 32–34 (in Russian). [4] A. Soleev, A.B. Aranson, Computation of a polyhedron and normal cones of its faces, Institute of Applied Mathematics, Preprint #36. Moscow, 1994 (in Russian). [5] T.S. Motzkin, G.J. Schoenberg, The relaxation method for linear inequalities, Can. J. Math. 6 (1954) 393–404. [6] E. Burger, Uber homogene lineare Ungleichungssysteme, Z. angew. Math. und Mech. 36 (34) (1956) 135–139. [7] A. Soleev, A.B. Aranson, First approximations of a reversible ODE system, Institute of Applied Mathematics, Preprint #28, Moscow, 1995 (in Russian). [8] A.D. Bruno, A. Soleev, The local analysis of singularities of a reversible ODE system, Trudy Mosk. Mat. Obsch. 59 (1998) 3–72 (in Russian); Transactions of Moscow Math. Soc. 59 (1998) (in English). [9] V.V. Vedenyapin, Yu.N. Orlov, On conservation laws for polynomial Hamiltonians and for discrete models of the Boltzman equation, Teoreticheskaja i Matematicheskaja Fizika, 121 (2) (1999) 307–315 (In Russian); Theoretical and Mathematical Physics, 121 (2) (1999) 1516–1523. [10] M. Blaug, Economic Theory in Retrospect, Cambridge University Press, 1985.
Computation and applications of the Newton polyhedrons Alexander B. Aranson Department of Mathematics, Keldysh Institute of Applied Mathematics, Miusskaja Sq. 4, Moscow 125047, Russia
Abstract We consider the multivariate Laurent polynomial f (X) = aQ X Q , Q ∈ D with coefficients aQ ∈ R or C and D is some set in Zn . The set D = D(f ) = {Q: aQ = 0} is called the support of the polynomial f (X). The convex hull M = M(f ) of the set D is called the Newton polyhedron of the polynomial f (X). There are important correspondences between properties of the polynomial f (X) and of its Newton polyhedron M(f ) that were studied by Bruno, Soleev, Khovansgfkii and others. We propose algorithms and the computer program for computation of the Newton polyhedron of any polynomial, and for computation of all elements of this polyhedron (vertices, edges, faces, etc.). © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. Keywords: Systems of differential polynomials; Newton polyhedron; Intersections of normal cones; Truncated systems
We consider the Laurent polynomial f (X) = aQ XQ , Q ∈ D
(1) q
q
q
where X = (x1 , x2 , . . . , xn ) ∈ Rn or Cn , Q = (q1 , q2 , . . . , qn ) ∈ Zn , monomial X Q = x1 1 x2 2 . . . xnn , coefficients aQ ∈ R or C and D is some set in Zn . The set D = D(f ) = {Q: aQ = 0} is called the support of the polynomial f (X). The convex hull M = M(f ) of the set D is called the Newton polyhedron of the polynomial f (X). The boundary ∂M of the polyhedron M consists of faces Γk(d) of different dimension d(0 ≤ d < n). Zero dimensional faces Γk(0) are vertices of the polyhedron M, one dimensional faces are its edges etc. Let Rn∗ denote the dual space such that for P = (p1 , . . . , pn ) ∈ Rn∗ and Q = (q1 , . . . , qn ) ∈ Rn with the scalar product P , Q = p1 q1 + . . . + pn qn . For a fixed vector P = 0, let DP denote such subset of the set D that in each point Q ∈ DP the scalar product P , Q has the maximal value cP among all Q ∈ D, that is P , Qj = cP
for Qj ∈ DP
and P , Qj < cP
for Qj ∈ D \ DP .
E-mail address: [email protected] (A.B. Aranson). 0378-4754/01/$ – see front matter © 2001 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 4 7 5 4 ( 0 1 ) 0 0 3 3 5 - 4
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A.B. Aranson / Mathematics and Computers in Simulation 57 (2001) 155–160
According to (2), to each vector P ∈ Rn∗ there corresponds a supporting to the set D hyperplane LP = {Q: P , Q = cP } lying in Rn . It intersects the polyhedron M along a face Γk(d) , where d is the dimension of the face and k is its number. The set Uk(d) of all vectors P ∈ Rn∗ such, that LP ∩ M = Γk(d) , is called the normal cone of the face Γk(d) . Let dim M = n. The normal cone Uk(n−1) of the hyperface Γk(n−1) is a ray orthogonal to the face and directed out of the polyhedron M. The normal cone Uk(n−2) is a sector bounded by rays Ul(n−1) and Um(n−1) if Γk(n−2) = Γl(n−1) ∩ Γm(n−1) etc. The union U of normal cones of all faces is Rn∗ \ {0}. Let Dk(d) = Γk(d) ∩ D be a boundary subset of the set D. Lemma 1. [1,2]. If P ∈ Uk(d) , then DP = Dk(d) . The sum fj XQj for Qj ∈ Dk(d) fˆk(d) (X) =
(3)
is called the truncation of the polynomial (1). Now we consider a curve of the form xi = bi τ pi (1 + o(1)),
bi = 0,
i = 1, . . . , n,
(4)
where τ → ∞. On such curve a monomial XQ = B Q τ P ,Q (1 + o(1)), where B = (b1 , . . . , bn ) and P = (p1 , . . . , pn ). If P ∈ Uk(d) , then on the curve (4), the polynomial (1) is f (X) = fˆk(d) (B)τ cP +τ cP o(1), where cP is from (2). Thus, the truncation (3) is the first approximation of the polynomial f (X) on curves (4) with P ∈ Uk(d) . If curve (4) is a solution of the equation f (X) = 0 then fˆk(d) (B) = 0. That is we have Theorem 1. [1,2]. Let the curve (4) be a solution of the equation f (X) = 0 and P ∈ Uk(d) . Then the first approximation xi = bi τ pi , i = 1, . . . , n of the solution (4) is a solution of the corresponding truncated equation fˆk(d) (X) = 0. This approach allows to find solutions of the equation f = 0 with any accuracy [1]. For other applications of the Newton polyhedron in various mathematical fields see in [1]. Example 1. Let n = 2 and f = x13 +x23 −2x1 x2 , then the set D = {Q1 , Q2 , Q3 } = {(3, 0), (0, 3), (1, 1)}. Its boundary subset are D1(0) = Q1 ,
D2(0) = Q3 ,
D1(1) = {Q1 , Q3 },
D3(0) = Q2
D2(1) = {Q2 , Q3 },
D3(1) = {Q1 , Q2 }
and some of their normal cones are U1(1) = −λ(1, 2),
U2(1) = −λ(2, 1),
U3(1) = −λ(1, 1)
(λ > 0).
The Newton polyhedron and normal cones see in Fig 1. Among truncations fˆk(d) there are fˆ1(0) = x13 ,
fˆ2(0) = −2x1 x2 ,
fˆ3(0) = x23 ;
fˆ1(1) = x13 − 2x1 x2 ,
fˆ2(1) = x23 − 2x1 x2 .
A.B. Aranson / Mathematics and Computers in Simulation 57 (2001) 155–160
157
Fig. 1. The Newton polyhedron (a) and normal cones of its faces (b) for Example 1.
Roots of the truncated equation fˆ1(1) = 0 are x2 = 1/2x12 , i.e. the first approximation of a branch of the f = 0 near X = (x1 , x2 ) = 0. The roots of the equation fˆ2(1) = 0 give the first approximation of another branch of the curve near the origin. The aim of the talk is to describe the algorithm and computer program for the computation of all boundary subsets Dk(d) for any given finite set D = {Q1 , . . . , Qm } ⊂ Zn and their normal cones Uk(d) . The system of equations and inequalities (2) determines two objects: the boundary subset DP and its normal cone UP . Thus, it is necessary to find solution of every such system (2) to select all DP and UP . But all these systems can be presented as the single system of linear weak inequalities P , Qj ≤ cP ,
j = 1, . . . , m.
(5)
˜ = (Q, qn+1 ) where additional Instead of vectors P and Q we consider vectors P˜ = (P , pn+1 ) and Q coordinate pn+1 = −cP and qn+1 = 1 [3]. Now the system (5) becomes the system of linear homogeneous weak inequalities ˜ j ≤ 0, P˜ , Q
j = 1, . . . , m.
(6)
The set of solutions to system (6) is a polyhedral convex cone U˜ ⊂ Rn+1 ∗ . The skeleton of the cone ˜ U˜ consists of some vectors P˜1 , . . . , P˜l ∈ Rn+1 , so the cone U is their conic hull. Each vector P˜k = ∗ (n−1) of the polyhedron M. Moreover to each (Pk , pk,n+1 ) where Pk is a normal vector to a hyperface Γk (d+1) ˜ ˜ of the cone U , there corresponds the normal cone Uk(d) of the face Γk(d) of the polyhedron face Uk M. In more details, let P˜k1 , . . . , P˜ks form the skeleton of the case U˜ k(d+1) ⊂ U˜ , then the boundary subset ˜ j = 0, i = 1, . . . , s}. Dk(d) = {Qj ∈ D: P˜ki , Q ˜ The vectors Pk of the skeleton of the cone U˜ can be obtained by Motzkin–Burger algorithm [1,4–6]. The algorithm consists of several steps. By the first step one obtains solutions of the first inequality, by the second step one take in account the second inequality etc. The geometry of this algorithm is the projection of skeleton vectors of the solutions of previous inequalities into the boundary of the half-space determined by the current inequality. ˜ 1 = (3, 0, −1), Q ˜ 2 = (0, 3, −1), Q ˜ 3 = (1, 1, −1). Example 2. (continuation of Example 1). Here Q Three steps of the Motzkin–Burger algorithm for the set D of Example 1 are shown in Figs. 2–4.
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˜ 1 ≤ 0. The vector Q ˜ 1 ∈ R3 , the vector P˜11 ∈ R3∗ and Fig. 2. Step 1 for Example 2. The set of solutions to the inequality P˜ , Q ˜ 1 , are shown. part of the hyperplane H in R3∗ , with a basis B˜ 11 , B˜ 21 , which is normal to the vector Q
˜ 1 ≤ 0. The set of its solutions consists of a half-space, Step 1: We are solving the inequality P˜ , Q 1 ˜ which is bounded by plane H with basis B1 = (0, 1, 0) and B˜ 21 = (1, 0, −3). The vector P˜11 = (−1, 0, 0) belongs to the interior of the half-space (see Fig. 2). Step 2: We are calculating the intersection of the half-space, found in the step 1, with the half-space, ˜ 2 ≤ 0. This intersection is a dihedral angle with the vertex in the form of defined by the inequality P˜ , Q the line L. The basis of this line is the vector B˜ 12 = (−1, −1, 3). The vector P˜12 = (1, −10, −3) belongs to the interior of one face of the dihedral angle and the vector P˜22 = (−10, 1, −3) belongs to another face (see Fig. 3). Step 3: We are calculating the intersection of the dihedral angle, found in the step 2, with the half-space, ˜ 3 ≤ 0. Vectors P˜1 = (−2, −1, 3), P˜2 = (−1, −2, 3), P˜3 = (1, 1, −3) defined by the inequality P˜ , Q
˜ 1 ≤ 0, P˜ , Q ˜ 2 ≤ 0. Vectors Q ˜ 1, Q ˜ 2 ∈ R3 , Fig. 3. Step 2 for Example 2. The set of solutions to the system of inequalities P˜ , Q ˜ 1, Q ˜ 2 , and dihedral angle of solutions are shown. vectors P˜12 , P˜22 ∈ R3∗ , the line L with a basis B˜ 12 , which is normal to vectors Q
A.B. Aranson / Mathematics and Computers in Simulation 57 (2001) 155–160
159
˜ 1 ≤ 0, P˜ , Q ˜ 2 ≤ 0, P˜ , Q ˜ 3 ≤ 0. Vectors Fig. 4. Step 3 for Example 2. The set of solutions to the system of inequalities P˜ , Q ˜ 1, Q ˜ 2, Q ˜ 3 ∈ R3 and vectors P˜1 , P˜2 , P˜3 ∈ R3∗ which form the skeleton of the cone of solutions, are shown. Q
˜ 1 ≤ 0, P˜ , Q ˜ 2 ≤ 0, form the skeleton of the cone, which is the set of solutions of the system P˜ , Q ˜ 3 ≤ 0 (see Fig. 4). P˜ , Q I have realized the described algorithms as a computer program in the C++ language [3,4]. This program reads file with coordinates of all points of the set D and calculate following items: • All normal vectors Pk of hyperfaces Γk(n−1) of the polyhedron M and constants cP . • The corresponding matrix where the columns of this matrix correspond to the vectors Qj of the set D and rows correspond to normal vectors Pk of hyperfaces Γk(n−1) of M. In the intersection of the column with number j and the row with number k, we write the sign “+” if Qj ∈ Dk(n−1) and the sign “−” in the contrary case (see Table 1). • The list of all boundary subsets Dk(d) . For each of them, the program writes its dimension d, numbers j of all points Qj ∈ Dk(d) and numbers l of normal vectors Pl forming the skeleton of the normal cone Uk(d) (see Table 2). The program was used in a study of surface waves on water (with n = 6) ([1], Ch. V; and [7,8]) and in a study of the discrete models of the Boltzman equation (with n = 25) [9]. The algorithm and the Table 1 The corresponding table for Example 1
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Table 2 The list of all boundary subsets Dk(d) for Example 1 k
d
j
l
1 2 3 1 2 3
1 1 1 0 0 0
1,3 2,3 1,2 1 2 3
1 2 3 1,3 2,3 1,2
computer program described above can be applied to other fields of mathematics, in particularly, for a selection of Pareto-optimal sets in the multicriterial optimization [10]. Acknowledgements The author thanks Prof. A.D. Bruno for his supervision of the work. The work was supported by RFBR, Grant 99-01-01063. References [1] A.D. Bruno, Power Geometry in Algebraic and Differential Equations, Fizmatgiz, Moscow, 1998, p. 288 (in Russian, its English translation was published by Elsevier, 2000). [2] A.D. Bruno, Power geometry, J. Dynamic. Contr. Syst. 3 (4) (1997) 471–491. [3] A.B. Aranson, Computation of the Newton polyhedron, in: Proceedings of the International Conference Devoted to 175th Anniversary of P.L. Chebyshev, Moscow State University, 1 (1996) 32–34 (in Russian). [4] A. Soleev, A.B. Aranson, Computation of a polyhedron and normal cones of its faces, Institute of Applied Mathematics, Preprint #36. Moscow, 1994 (in Russian). [5] T.S. Motzkin, G.J. Schoenberg, The relaxation method for linear inequalities, Can. J. Math. 6 (1954) 393–404. [6] E. Burger, Uber homogene lineare Ungleichungssysteme, Z. angew. Math. und Mech. 36 (34) (1956) 135–139. [7] A. Soleev, A.B. Aranson, First approximations of a reversible ODE system, Institute of Applied Mathematics, Preprint #28, Moscow, 1995 (in Russian). [8] A.D. Bruno, A. Soleev, The local analysis of singularities of a reversible ODE system, Trudy Mosk. Mat. Obsch. 59 (1998) 3–72 (in Russian); Transactions of Moscow Math. Soc. 59 (1998) (in English). [9] V.V. Vedenyapin, Yu.N. Orlov, On conservation laws for polynomial Hamiltonians and for discrete models of the Boltzman equation, Teoreticheskaja i Matematicheskaja Fizika, 121 (2) (1999) 307–315 (In Russian); Theoretical and Mathematical Physics, 121 (2) (1999) 1516–1523. [10] M. Blaug, Economic Theory in Retrospect, Cambridge University Press, 1985.