Newton Polygons of Polynomial Ordinary Differential Equations

Transcription

1 Int. Journal of Math. Analysis, Vol. 5, 2011, no. 35, Newton Polygons of Polynomial Ordinary Differential Equations Ali Ayad 1 Lebanese university, Faculty of sciences Section 1, Hadath, Beirut, Lebanon ayadali99100@hotmail.com Abstract In this paper we show some properties of the Newton polygon of a polynomial ordinary differential equation. We give the relation between the Newton polygons of a differential polynomial and its partial derivatives. Newton polygons of evaluations of differential polynomials are also described. 1 Introduction Newton polygons construct a useful tool for solving polynomial ordinary differential equations. Earlier, Briot and Bouquet [1] used Newton polygon methods for solving first order and first degree ordinary differential equations. Recently, Grigoriev and Singer [8], Cano [2, 3] and Della Dora et. al. [6] described Newton polygon algorithms for computing Puiseux series and series with real exponents as solutions of polynomial differential equations. Newton polygons are also useful for computing Newton-Puiseux expansions for the roots of polynomials and for factoring polynomials over fields of formal power series [4, 5]. In this paper, we describe some useful properties of Newton polygons of differential polynomials. Section 2 gives the definition of the Newton polygon of a polynomial ordinary differential equation. Section 3 describes the relation between the Newton polygons of a differential polynomial and its partial derivatives. Section 4 establishes Newton polygon of different evaluations of differential polynomials. Let K be a field and K be an algebraic closure of K. Let L and L be the two following fields: L = ν N Kx 1 ν, L = ν N Kx 1 ν 1 We gratefully thank Professor Dimitri Grigoriev for his help in the redaction of this paper, and more generally for his suggestions about the approach presented here.

2 1712 A. Ayad which are the fields of fraction-power series of x over K resp. K, i.e., the fields of Puiseux series of x with coefficients in K resp. K. Each element ψ L resp. ψ L can be represented in the form ψ = c i Q ix i, c i K resp. c i K. The order of ψ is defined by ordψ := min{i Q,c i 0}. The fields L and L are differential fields with the differentiation operator d dx ψ = ic i x i 1. i Q Let y 0,...,y n be new variables algebraically independent over K and F y 0,...,y n be a polynomial in the variables y 0,...,y n with coefficients in L. This polynomial defines an ordinary differential equation F y, dy,..., dn y = 0 which will dx dx be denoted by F y = 0. We can write F in the form: F = f i,α x i y α 0 0 yαn n, f i,α K i Q,α A where α =α 0,...,α n belongs to a finite subset A of N n+1. The order of F is defined by ordf := min {i Q; f i,α 0 for a certain α}. We define the degree of F w.r.t. x by deg x F = max {i Q; f i,α 0 for a certain α} it can be equal to +. 2 Newton polygons of polynomial differential equations We define the Newton polygon of F as follow. For every couple i, α Q A such that f i,α 0 i.e., every existing term in F, we mark the point P i,α := i α 1 2α 2 nα n,α 0 + α α n Q N. We denote by P F the set of all the points P i,α. The convex hull of these points and the point +, 0 in the plane R 2 is denoted by N F and is called the Newton polygon of the differential equation F y = 0 in the neighborhood of x = 0. If deg y0,...,y n F =m, then N F is located between the two horizontal lines y = 0 and y = m. For each a, b Q 2 \{0, 0}, we define the set NF, a, b :={u, v P F, u,v P F, au + bv au + bv}. A point P i,α P F is a vertex of the Newton polygon N F if there exist a, b Q 2 \{0, 0} such that NF, a, b ={P i,α }. We remark that N F has a finite number of vertices. A pair of different vertices e =P i,α,p i,α forms an edge of N F if there exist a, b Q 2 \{0, 0} such that e NF, a, b. We denote by EF resp. V F the set of all the edges e resp. all the vertices

3 Newton polygons of polynomial ODE 1713 p ofn F for which a>0 and b 0 in the previous definitions. It is easy to prove that if e EF, then there exists a unique pair ae,be Z 2 such that GCDae,be = 1, ae > 0, be 0 and e NF, ae,be where GCD is an abbreviation of Greatest Common Divisor. By the inclination of a line we mean the negative inverse of its geometric slope. If e EF, we can prove that the fraction μ e = be Q is the inclination of the straight line ae passing through the edge e. If p V F and NF, a, b ={p} for a certain a, b Q 2 \{0, 0}, then the fraction μ = b Q is the inclination of a a straight line which intersects N F exactly in the vertex p. For each edge e EF, we define the univariate polynomial in a new variable Z H F,e Z = P i,α NF,ae,be f i,α Z α 0+α 1 + +α n μ e α 1 1 μ e αn n K[Z], where μ e k := μ e μ e 1 μ e k + 1 for any positive integer k. We call H F,e Z the characteristic polynomial of F associated to the edge e EF. Its degree is at most m = deg y0,...,y n F. If ψ L is a solution of the differential equation F y = 0 such that ordψ =μ, i.e., ψ has the form ψ = i Q,i μ c ix i, c i K, then there exists an edge e EF such that μ e = μ and H F,e c μ = 0, i.e. c μ is a root of the polynomial H F,e in K. This condition is called a necessary initial condition to have a solution of F y = 0 in the form of ψ see Lemma 1 of [3]. Namely, H F,e c μ is equal to the coefficient of the lowest term in the expansion of F ψx with indeterminates μ and c μ. For each vertex p =u, v V F, let μ 1 <μ 2 be the inclinations of the adjacent edges at p in N F. It is easy to prove that for all rational number μ = b a, a N, b N such that NF, a, b ={p}, we have μ 1 <μ<μ 2. We associate to p the polynomial h F,p μ = P i,α =p f i,α μ α 1 1 μαn n K[μ], which is called the indicial polynomial of F associated to the vertex p here μ is considered as an indeterminate. Let H F,p Z =Z v h F,p μ defined as above for edges e EF. Remark 2.1 Let p =u, v V F and e be the edge of N F descending from p, then h F,p μ e is the coefficient of the monomial Z v in the expansion of the characteristic polynomial of F associated to e.

4 1714 A. Ayad 3 Newton polygons of partial derivatives of differential polynomials Write F in the form F = F F m where m = deg y0,...,y n F and F s = i Q, α =s f i,αx i y α 0 0 yn αn is the homogeneous part of F of degree s with respect to the indeterminates y 0,...,y n, α = α 0,...,α n A and α = α α n is the norm of α. Then the ordinate of any point of P F s is equal to s and P F = 0 s m P F s. Let 0 j n. If there exists an integer k 1 such that for all 1 s m such that F s 0, D s,j := deg yj F s k then we can easily prove that P k F yj k is the translation of P F defined by the point kj, k, i.e., k F k F P = P F +{kj, k} and then N = N F +{kj, k}. yj k yj k For any a, b Q 2 \{0, 0}, we have k F N Thus the edges of N each e E k F y k j H k F,e Z = y j k For each p V y k j k F y k j,a,b = NF, a, b+{kj, k}. are exactly the translation of those of N F. For, its characteristic polynomial is P i,α NF,ae,be k F y k j f i,α Z α 0+α 1 + +α n k α j k μ e α 1 1 μ e α j k j μ e αn n K[Z]., its indicial polynomial is h k F,p μ = f i,α α j k μ e α 1 1 μ e α j k j μ e αn n K[μ]. y j k P i,α =p Let 0 n. If there exist integers k 1,k 2 1 such that for all 1 s m, D s,j2 k 2 and deg k 2 F s yj1 k 1 then and then k 1 +k 2 F P = P F +{k 1 + k 2, k 1 k 2 } k 1 +k 2 F N = N F +{k 1 + k 2, k 1 k 2 }.

5 Newton polygons of polynomial ODE 1715 For any a, b Q 2 \{0, 0}, we have k 1 +k 2 F N = NF, a, b+{k 1 + k 2, k 1 k 2 }. For each e E k 1 +k 2F P i,α NF,ae,be, its characteristic polynomial is H k 1 +k 2 F,e Z = f i,α Z α 0+α 1 + +α n k 1 k 2 α j1 k1 α j2 k2 μ e α 1 1 μ e α k 1 μ e α k 2 μ e αn n. Theorem 3.1 Let k 1 be an integer such that for all 1 s m such that F s 0 and for all 0 j n we have D s,j k. For any e EF, the k-th derivative of H F,e K[Z] is given by the formula H k F,e Z = 0 k 0,...,k n n μ e k 1 1 μ e kn n H k F y k 0 0 ykn n,e Z. where the sum ranges over all the partitions k 0,...,k n of k, i.e., k 0 + +k n = k. Proof. By induction on k taking into account the above discussion. 4 Newton polygons of evaluations of differential polynomials Let F be a differential polynomial as in Section 1. Let 0 c K, μ Q and Gy =F cx μ +y be the differential polynomial obtained from F by replacing y k by cμ k x μ k + y k for all 0 k n where μ 0 := 1. In this section, we will construct the Newton polygon of the differential equation Gy = 0 for different values of c and μ. For each differential monomial my =f i,α x i y α 0 0 yαn n of F with corresponding point p P F, compute mcx μ + y =f i,α x i cx μ + y 0 α 0 cμx μ 1 + y 1 α1 cμ n x μ n + y n αn. Remark that the corresponding points of the differential monomials of mcx μ + y have ordinate less or equal than s = α 0 + α α n and lie in the line passing through p with inclination μ. There are two possibilities for μ:

6 1716 A. Ayad Theorem 4.1 If μ = μ e is the inclination of an edge e EF. For any 0 s m = deg y0,...,y n F, the vertex of N G of ordinate s corresponds to the differential monomial of G with coefficient equals to q s c, μ e := 0 k 0 k n n 0 k 0 k n n 1 k 0! k n! H s F y k 0 0 ykn n,e c. where the sum ranges over all the partitions k 0,...,k n of s i.e., k 0 + +k n = s. Its x-coordinate is the minimum of the quantities i + μα α n s α 1 2α 2 nα n for i Q and α A. If μ 1 <μ<μ 2 where μ 1 and μ 2 are the inclinations of the two adjacent edges of a vertex p =u, v V F, then for any 0 s m, the vertex of N G of ordinate s corresponds to the differential monomial of G with coefficient equals to c v s 1 k 0! k n! h s F,p μ. y k 0 0 ykn n where the sum ranges over all the partitions k 0,...,k n of s. Proof. Let μ = μ e for e EF and compute Gy = f i,α x i cx μ + y 0 α 0 cμx μ 1 + y 1 α1 cμ n x μ n + y n αn. i Q,α A For each 0 s m, compute G s the homogeneous part of G of degree s in y 0,...,y n. We remark that for fixed i and α, all the differential monomials of G s have the same corresponding point which is i + μα α n s α 1 2α 2 nα n,s. The x-coordinate of the vertex of N G of ordinate s is the minimum of the x-coordinates i+μα 0 + +α n s α 1 2α 2 nα n for i Q and α A. This minimum is realized by the points P i,α NF, ae,be. This proves the lemma taking into account the formula for the characteristic polynomial of the derivatives of F in Section 3 see Theorem 3.1. The following Corollary is a generalization of Lemma 2.2 of [7] which deals with the Newton polygon of the Riccatti equation associated to a linear ordinary differential equation. Corollary 4.2 Let μ = μ e be the inclination of an edge e EF. The edges of N G, situated above the edge e are the same as in N F. Let an integer s 1 0 be such that q s c, μ e =0for all 0 s<s 1 and q s1 c, μ e 0 then N G has a vertex of ordinate s 1 and it has at least s 1 edges with inclination greater than μ e.

7 Newton polygons of polynomial ODE 1717 Proof. Let p s1 be the vertex of N G of ordinate s 1 and x-coordinate x ps1 the minimum of the values i+μα 0 + +α n s 1 α 1 2α 2 nα n for i Q and α A by Theorem 4.1. We have q s1 1c, μ e = 0, then the x-coordinate of the vertex p s1 1 of ordinate s 1 1 is strictly less than the minimum of the values i + μα α n s 1 +1 α 1 2α 2 nα n for i Q and α A. Thus the inclination of the edge joining p s1 and p s1 1 is greater than μ. Corollary 4.3 Let μ = μ e be the inclination of an edge e EF. If H F,e c =0then the intersection point of the straight line passing through e with the x-axis is not a vertex of N G and N G has an edge with inclination greater than μ e. Proof. We have q 0 c, μ e = H F,e c = 0, then s 1 1. This proves the corollary by applying Corollary 4.2. References [1] Ch. Briot and J.-C. Bouquet, Intégration des équations différentielles au moyen des fonctions elliptiques, J. École Impériale Polytechnique, , [2] J. Cano, On the series defined by differential equations, with an extension of the Puiseux Polygon construction to these equations, International Mathematical Journal of Analysis and its Applications, , [3] J. Cano, The Newton Polygon Method for Differential Equations, Computer Algebra and Geometric Algebra with Applications, 2005, [4] A. Chistov, Polynomial Complexity of the Newton-Puiseux Algorithm, Mathematical Foundations of Computer Science 1986, [5] A.L. Chistov, Polynomial complexity algorithms for computational problems in the theory of algebraic curves, Journal of Mathematical Sciences, , [6] J. Della Dora and F. Richard-Jung, About the Newton algorithm for nonlinear ordinary differential equations, Proceedings of the 1997 international symposium on Symbolic and algebraic computation, United States, [7] D. Grigoriev, Complexity of factoring and GCD calculating of ordinary linear differential operators, J. Symp. Comput., , 7-37.

8 1718 A. Ayad [8] D. Grigoriev and M. Singer, Solving ordinary differential equations in terms of series with real exponents, Trans. AMS, , Received: February, 2011