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Integral D-Finite Functions Integral D-Finite Functions ∗ y Manuel Kauers Christoph Koutschan RISC / Johannes Kepler University RICAM / Austrian Academy of Sciences 4040 Linz, Austria 4040 Linz, Austria [email protected] [email protected] ABSTRACT combination of 1; α; α2. An integral basis in this example is f1; α; 1 α2g. We propose a differential analog of the notion of integral 2 closure of algebraic function fields. We present an algorithm The concept of integral closure has been studied in rather for computing the integral closure of the algebra defined general domains [9,6]. To compute an integral basis for an by a linear differential operator. Our algorithm is a direct algebraic number field, special algorithms have been devel- analog of van Hoeij's algorithm for computing integral bases oped [7,5]. At least two different approaches are known of algebraic function fields. for algebraic function fields, i.e., the case when R = C[x] for some field C, k = C(x), and K = k[y]=hMi for some irreducible polynomial M 2 k[y]. The algorithm derived by Categories and Subject Descriptors Trager [10] in his thesis is an adaption of an algorithm for I.1.2 [Computing Methodologies]: Symbolic and Alge- number fields, and the algorithm by van Hoeij [12] is based braic Manipulation|Algorithms on the idea of successively canceling lower order terms of Puiseux series. General Terms The theory of algebraic functions parallels in many ways the theory of D-finite functions, i.e., the theory of solutions Algorithms of linear differential operators. It is therefore natural to ask what corresponds to the notion of integrality in this latter Keywords theory. In the present paper, we propose such a definition Integral Basis, D-finite Function, Differential Operator and give an algorithm which computes integral bases accord- ing to this definition. Our algorithm and the arguments un- derlying its correctness are remarkably similar to van Hoeij's 1. INTRODUCTION algorithm for computing integral bases of algebraic function The notion of integrality is a classical concept in the the- fields. ory of algebraic field extensions. If R is an integral domain, k In view of the key role that integral bases play for in- the quotient field of R, and K = k(α) an algebraic extension definite integration (Hermite reduction) of algebraic func- of k of degree d, then an element of K is called integral if its tions [10,3,2], we have hope that results presented below monic minimal polynomial has coefficients in R. While K will help to develop new algorithms for indefinite integration forms a k-vector space of dimension d, the set of all integral of D-finite functions. An example pointing in this direction elements of K forms an R-module, called the integral clo- is given in the end. sure (or normalization) of R in K, and commonly denoted by OK .A k-vector space basis of K which at the same time Acknowledgment. We want to thank the anonymous ref- generates OK as an R-module is called an integral basis. For erees for their detailed and valuable comments. p3 example, when R = Z, k = Q, and K = Q(α) with α = 4, 2 then the canonical vector space basisp f1; α; α g of K is not 1 2 3 an integral basis, because 2 α = 2 is an integral element 2. INTEGRAL FUNCTIONS, 3 of K (its minimal polynomial is x − 2) but not a Z-linear INTEGRAL CLOSURE, AND ∗Supported by the Austrian Science Fund (FWF): Y464. INTEGRAL BASES y Supported by the Austrian Science Fund (FWF): W1214. Throughout this paper, let C be a computable field of characteristic zero, C¯ an algebraically closed field contain- Permission to make digital or hard copies of all or part of this work for personal or ing C (not necessarily the smallest), and x transcendental classroom use is granted without fee provided that copies are not made or distributed ¯ ¯ for profit or commercial advantage and that copies bear this notice and the full cita- over C. When R is a subring of C(x), we write R[D] for the tion on the first page. Copyrights for components of this work owned by others than algebra of differential operators with coefficients in R, i.e., r ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or re- the algebra of all (formal) polynomials `0 + `1D + ··· + `rD publish, to post on servers or to redistribute to lists, requires prior specific permission with `0; : : : ; `r 2 R. This algebra is equipped with the natu- and/or a fee. Request permissions from [email protected]. ral addition and the unique noncommutative multiplication ISSAC’15, July 6–9, 2015, Bath, United Kingdom respecting the commutation rules Dc = cD for all c 2 R \ C¯ Copyright is held by the owner/author(s). Publication rights licensed to ACM. and Dx = xD + 1. Typical choices of R will be C[x], C¯[x], ACM 978-1-4503-3435-8/15/07 ...$15.00. ¯ DOI: http://dx.doi.org/10.1145/2755996.2756658 C(x), or C(x) in the following. r ¯ For an operator L = `0 + `1D + ··· + `rD 2 C[x][D] The function ι(·; j) specifies for each element ν +Z of C=Z ν j with `r 6= 0 we denote by ord(L) = r the order of L. Recall the smallest element ν such that x log(x) should be consid- ν j ν+1 j that such an operator with x - `r admits a fundamental sys- ered integral. If ι(ν+Z; j) = ν, then x log(x) ; x log(x) , tem of formal power series solutions, i.e., the vector space ::: are integral and xν−1 log(x)j ; xν−2 log(x)j ;::: are not. V ⊆ C¯[[x]] consisting of all the power series f with L · f = 0 The condition ι(Z; 0) = 0 implies that formal Laurent series has dimension r. When x j `r 6= 0, there is still a fun- are integral if and only if they are in fact formal power series. damental system of generalized series solutions of the form −1=s ν 1=s Example 2. A natural choice for C ⊆ is perhaps ι(z + exp(p(x ))x a(x ; log(x)) for some s 2 N, p 2 C¯[x], C ν 2 C¯, a 2 C¯[[x]][y]. (This notation is not meant to imply Z; 0) = z for all z 2 C with 0 ≤ <(z) < 1, and ι(z+Z; j) = z that a has a nonzero constant term, so the series in general for all z 2 C with 0 < <(z) ≤ 1 when j ≥ 1. With this ν ν+i convention, a term xν log(x)j is integral if and only if the does not start at x but at some x where i 2 N is such i that x is the lowest order term of a.) We restrict our at- corresponding function is boundedp in a small neighborhood of the origin. For example, 1, x −1, x log(x) all are integral, tention here to the case where p = 0 and s = 1. Moreover, p we want to assume that ν 2 C (this can always be achieved while x−1, x −1−1, log(x) are not. Unless otherwise stated, by a suitable choice of C), to ensure that the output of our we shall always assume this choice of ι in the examples given algorithm involves only coefficients in C. Hence we only below. consider operators L which admit a fundamental system in ¯ ¯ S xν C¯[[x]][log x]. It is well known [8] how to determine Proposition 3. Let α 2 C and let R be the set of all C- ν2C ν ¯ the first terms of a basis of such solutions for a given oper- linear combinations of series in (x−α) C[[x−α]][log(x−α)], ator L 2 C¯[x][D]. By a linear change of variables, the same ν 2 C. Then: techniques can also be used to find the first terms of solu- 1. In every series f 2 R there are at most finitely many tions in S (x − α)ν C¯[[x − α]][log(x − α)], for any given ν2C terms (x − α)µ log(x − α)j which are not integral. α 2 C¯. More precisely, if L belongs only to C[x][D] and α 2 C¯, then these solutions are actually linear combinations 2. The set R together with the natural addition and mul- S ν of elements of ν2C (x − α) C(α)[[x − α]][log(x − α)]. For a tiplication forms a ring, and f f 2 R j f is integral g field K with C ⊆ K ⊆ C¯ we will use the notation forms a subring of R. [ ν K[[[x − α]]] := (x − α) K[[x − α]][log(x − α)]: Proof. 1. First consider the case when f 2 (x − α)ν C¯[[x − ν2C α]][log(x − α)] for some ν 2 C. Let deg(f) denote the Observe that this is not a ring or a K-vector space. Also highest power of log(x − α) in f. Then the only possible observe that the exponents ν are restricted to the small non-integral terms in f can be (x − α)ν+i log(x − α)j for field C ⊆ K, although the dependence on the choice of C j 2 f0;:::; deg(f)g and i 2 f0; : : : ; ι(ν + Z; j) − ν − 1g.
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