Liouville theorem with parameters: asymptotics of certain rational integrals in differential fields
aa r X i v : . [ m a t h . C A ] A p r LIOUVILLE THEOREM WITH PARAMETERS:ASYMPTOTICS OF CERTAIN RATIONAL INTEGRALSIN DIFFERENTIAL FIELDS
MA LGORZATA STAWISKA
Abstract.
We study asymptotics of integrals of certain rationalfunctions that depend on parameters in a field K of characteristiczero. We use formal power series to represent the integral and provecertain identities about its coefficients following from generalizedVandermonde determinant expansion. Our result can be viewed asa parametric version of a classical theorem of Liouville. We alsogive applications. Integrating rational functions over differentialfields
Let D be a differential field. This means that D is a field with anadditional mapping ′ : D 7→ D (differentiation) satisfying the follow-ing two conditions: ( u + v ) ′ = u ′ + v ′ and ( uv ) ′ = u ′ v + uv ′ for all u, v ∈ D . The set K = { u ∈ D : u ′ = 0 } is a subfield of D , calledthe field of constants. We use the following terminology, adapted from[Ris]: Let U be a universal (differential) extension of D . For u ∈ U , u and D ( u ) are said to be simple elementary over D iff one of the fol-lowing conditions holds: (1) u is algebraic over D ; (2) There is a v in D , v = 0 such that v ′ = vu ′ (we will write equivalently u = log v ); (3)There is a v in D , v = 0 such that u ′ = uv ′ . (or equivalently u = exp v ).We say that F and any w ∈ F is elementary over D if F = D ( u , ..., u n )for some n , where each u i is simple elementary over D ( u , ..., u i − ) , i =1 , ..., n .The following theorem dates back to J. Liouville (cf. [Ris]): Mathematics Subject Classification.
Key words and phrases.
Liouville theorem, differential fields, integrals of rationalfunctions.
Theorem 1.
Let D be a differential field, F elementary over D . Sup-pose D and F have the same constant field K. Let g ∈ F , f ∈ D with g ′ = f . Then g = v + P c i log v i , where v , v i are elements of D and c i are elements of K. We will write f = g as equivalent to g ′ = f . Example and notation:
Let K be an arbitrary field of character-istic zero and z be transcendental over K . We introduce differentiationin the polynomial ring K [ z ] by taking z ′ = 1 and a ′ = 0 for all a ∈ K (the standard differentiation of polynomials in one variable). The field K ( z ) of rational fractions of K [ z ] is a differential field when we extend( z − ) ′ = − z − z − , and K is its field of constants.With K as in the example, we will consider the ring K [[1 /z ] of thefollowing formal series: P ∞ n =0 a n z − n with a n ∈ K . The differentiation ’can be extended term-by-term as a map of K [[1 /z ]] to itself. One canalso define a valuation o : K [[1 /z ]] N ∪ {∞} as follows (cf. [VS],discussion before Proposition 2.3.16): o ( f ) = min { n : a n = 0 } and o (0) = ∞ .Consider a a square-free polynomial Q ( z ) = z ( z − a ) ... ( z − a q ) with a n ∈ K . Our result can be now formulated as follows: Theorem 2. (a) In an elementary field F over K ( z ) , consider theelements g = R f ∈ F with f = 1 /Q , where Q ( z ) = z ( z − a ) ... ( z − a q ) is a square-free polynomial with a , ..., a q ∈ K, q ≥ . The set G of all such elements is in a bijective correspondence with a subset of K [[1 /z ]] .(b) For (the image of ) a g = R /Q we have o ( g ) = q , where q =deg Q + 1 . In the proof of this theorem we will apply the following identities:
Lemma 1. Q ′ (0) + 1 Q ′ ( a ) + ... + 1 Q ′ ( a q ) = 0 ,a Q ′ ( a ) + ... + a q Q ′ ( a q ) = 0 , ... a q − Q ′ ( a ) + ... + a q − q Q ′ ( a q ) = 0 , IOUVILLE THEOREM WITH PARAMETERS 3 a q Q ′ ( a ) + ... + a qq Q ′ ( a q ) = 1 ,a q + l Q ′ ( a ) + ... + a q + lq Q ′ ( a q ) = S l ( a , ..., a q ) , where S l is the complete homogeneous polynomial of degree l , sym-metric in its variables, i.e., S l ( X , ..., X n ) = P ≤ i ≤ ... ≤ i l ≤ n X i ...X i l for l = 1 , , ... . Proof. (of Lemma) We use properties of the Vandermonde determinant: V n ( x , ..., x n ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x . . . x n − x . . . x n − . . . . . . . . . . . . . . . . x n . . . x n − n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Recall that V n ( x , ..., x n ) = Q ≤ i 1) minus its Laplace expansion alongthe column of 1’s . In the sum of a ki Q ′ ( a i ) , k ≤ q − 1, the numerator isthe Laplace expansion of (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ...a q − . . . . . . . . . . . . . . . . a n ...a q − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) MA LGORZATA STAWISKA along the column containing terms of the type a k , and in the sumof a q + li Q ′ ( a i ) the numerator is the Laplace expansion of V q,l ( a , ...a q ) withrespect to the last column, which is a product of V q ( a , ...a q ) by thecomplete homogeneous symmetric polynomial S l ( a , ...a q ) of degree l > (cid:3) Remark . The last identity was obtained in a different way as Theorem3.2 in [Co], where it is also traced back to C.G.J. Jacobi. Proof. (of Theorem 2)First note that for a ∈ K we can identify ( z − a ) − with P ∞ n =1 a n z − ( n +1) .More generally, if Q ( z ) = z ( z − a ) ... ( z − a q ) is a square-free polyno-mial with a n ∈ K and P ∈ K [ z ], then partial fraction decomposi-tion gives P/Q = c /z + c / ( z − a ) + ... + c q / ( z − a q ) with c n = P ( a n ) /Q ′ ( a n ) , n = 1 , ..., q (cf. [Tr]) and P/Q can also be identifiedwith an element of K [[1 /z ]]. It follows that log(1 − a/z ) can be identi-fied with P ∞ n =1 (( − n a n /n ) z − n . Let now g = R (1 /Q ) with Q as above.Then g = b + Q ′ (0) log z + Q ′ ( a ) log( z − a ) + ... + Q ′ ( a q ) log( z − a q )in F with b in K . Identifying each log( z − a j ) , j = 1 , ..., q withan appropriate formal series in K [[1 /z ]] as above and adding the re-sults, we get g = P ∞ n =0 b n z − n . By the lemma, b = ... = b q − = 0, b q = 1 /q and b q + l = S l ( a , ..., a q ) / ( q + 1), where S l is the completehomogeneous symmetric polynomial of degree l , l = 1 , , ... . To en-sure this is the only possible series in K [[1 /z ]] that can be identifiedwith g = R (1 /Q ), note that any such series should be symmetric withrespect to a , ..., a q . Theorem 9.3 in [LS] says the following: If a for-mal series F in the variables X , ..., X q , Y is symmetric with respectto ( X , ..., X q ), then F = Φ( σ , ..., σ q , Y ), where Φ is a formal seriesin the variables X , ..., X q , Y and σ , ..., σ q are elementary symmetricpolynomials in X , ..., X q . Moreover, the series Φ is unique. Uniquenessof our P ∞ n =0 b n z − n follows, because S l ( X , ..., X q ) can be expressed aspolynomials in σ , ..., σ q . Hence also o ( g ) = q . (cid:3) Applications A particular case of our Theorem 2 is Proposition 1 in [GS], whichwas proved for K = C and a j = ζ j a , j = 1 , ..., q , where ζ is a primitiveroot of unity of order q . The convergence of R / ( z ( z − a ) ... ( z − a q )) →− / ( qz q ) as a , ..., a q → | z | > R ) isimportant in constructing approximate Fatou coordinates for analyticmaps f in a neighborhood of an f ( z ) = z + z q +1 + ... with q > f looks like a translation. The first IOUVILLE THEOREM WITH PARAMETERS 5 step in constructing Fatou coordinate for f consists in lifting f toa neighborhood of infinity by the coordinate change z 7→ − / ( qz q ).We considered f belonging to an one-parameter family of polynomi-als P λ ( z ) = λz + z with λ = e πip/q and λ = e πi ( p/q + u ) , with p, q coprime integers and u in a sufficiently small neighborhood of 0 in C .We started the construction of near-Fatou coordinate by applying thetransformation w ( z ) = R zz (1 /Q ( u, ζ )) dζ , where Q ( u, z ) is the Weier-strass polynomial for P ◦ qλ ( z ) − z . As u is small, the non-zero solutionsof Q ( u, z ) = 0 are also small. Because of convergence of integrals, thecoordinates obtained for P λ depend continuously on u . For more de-tails and references see [GS].Another application is a generalization of the well-known formula forelectrostatic potential of a dipole located at z = 0 (cf. [Ne]): considertwo charges 1 /a and − /a placed respectively at z = 0 and z = a . Thentheir combined electrostatic potential is (1 /a ) log z − (1 /a ) log( z − a ),which tends to − /z as a → /Q ′ (0) , /Q ′ ( a ) , ..., /Q ′ ( a q ) placed at 0 , a , ..., a q (with Q ( z ) = z ( z − a ) ... ( z − a q )), and it follows from Theorem 2 thatthe potential tends to − / ( qz q ) as a , ..., a q → Acknowledgment: The main idea of the paper occured while Iwas working with Estela Gavosto on [GS]. I thank her for many usefulconversations. References Analiza formalna i funkcje analityczne ,Wydawnictwo Uniwersytetu Jagiello´nskiego [ Formal Analysis and AnalyticFunctions , Jagiellonian University Press], Krak´ow, 2005[Ma] Macdonald, I. G.: Symmetric functions and Hall polynomials. Second edition.With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Ox-ford Science Publications. The Clarendon Press, Oxford University Press,New York, 1995.[Ne] Needham, Tristan: Visual complex analysis. The Clarendon Press, OxfordUniversity Press, New York, 1997[Ris] Risch, Robert H.: The problem of integration in finite terms. Trans. Amer.Math. Soc. 139 (1969), 167–189 MA LGORZATA STAWISKA [VS] Villa Salvador, Gabriel Daniel: Topics in the theory of algebraic functionfields. Mathematics: Theory & Applications. Birkh¨auser Boston, Inc., Boston,MA, 2006.[Tr] B. Trager: Algebraic factoring and rational function integration, Proceed-ings of the 1976 ACM Symposium on Symbolic and Algebraic Computation ,manuscript Department of Mathematics, University of Kansas, Lawrence, KS66045 E-mail address ::