For given multivariate functions specified by algebraic, differential or difference equations,the separability problem is to decide whether they satisfy linear differential or difference equations in one variable. In this paper, we will explain how separability problems arise naturally in creative telescoping and present some criteria for testing the separability for several classes of special functions,including rational functions, hyperexponential functions, hypergeometric terms, and algebraic functions.

Parallel telescoping is a natural generalization of differential creative-telescoping for single integrals to line integrals. It computes a linear ordinary differential operator L, called a parallel telescoper, for several multivariate functions, such that the application of L to the functions yields partial derivatives of a single function. We present a necessary and sufficient condition guaranteeing the existence of parallel telescopers for differentially finite functions, and develop an algorithm to compute minimal ones for compatible hyperexponential functions. Besides computing annihilators of parametric line integrals, we use the parallel telescoping for determining Galois groups of parameterized partial differential systems of first order.

Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. An order-degree curve for a given Ore operator is a curve in the (r,d)-plane such that for all points (r,d) above this curve, there exists a left multiple of order r and degree d of the given operator. We give a new proof of a desingularization result by Abramov and van Hoeij for the shift case, and show how desingularization implies order-degree curves which are extremely accurate in examples.

We show that the problem of constructing telescopers for rational functions of m + 1 variables is equivalent to the problem of constructing telescopers for algebraic functions of m variables and we present a new algorithm to construct telescopers for algebraic functions of two variables. These considerations are based on analyzing the residues of the input. According to experiments, the resulting algorithm for rational functions of three variables is faster than known algorithms, at least in some examples of combinatorial interest. The algorithm for algebraic functions implies a new bound on the order of the telescopers.

We present a method for determining the one-dimensional submodules of a Laurent-Ore module. The method is based on a correspondence between hyperexponential solutions of associated systems and one-dimensional submodules. The hyperexponential solutions are computed recursively by solving a sequence of first-order ordinary matrix equations. As the recursion proceeds, the matrix equations will have constant coefficients with respect to the operators that have been considered.

In this paper we show how to compute the Galois group G of a polynomial ƒ ∈ Q(x)[Y] by factoring the associated linear differential equation Lƒ(Y) = 0 (and constructions of it) of minimal order satisfied by the roots of ƒ. We use that the differential Galois group of Lƒ(Y) is a faithful linear representation of G whose character is a summand of the permutation character of G acting on the roots of ƒ. Our approach is motivated by the fact that the orders of the involved differential equations are much lower than the degrees of the Lagrange resolvants of ƒ. In the final section we show how, if ƒ ∈ Q(x)[Y], our approach via differential Galois theory helps one to also compute the Galois group of ƒ over Q(x).

Let $\rho$ be a set of algebraic expressions constructed with radicals and arithmetic operations, and which generate the splitting field F of some polynomial. Let $N_{\beta}(\rho)$ be the minimum total number of root-takings and exponentiations used in any straightline program for computing the functions in $\rho$ by taking roots, exponentials, logarithms, and performing arithmetic operations. In this paper it is proved that $N_{\beta}(\rho) = v(G)$, where $v(G)$ is the minimum length of any cyclic Jordan-Hölder tower for the Galois group G of F. This generalizes a result of Ja’Ja’ [Proceedings of the 22nd IEEE Symposium on Foundations of Computer Science, 1981, pp. 95–100], and shows that the inclusion of certain new primitives, such as taking exponentials and logarithms, does not improve the cost of computing such expressions as compared with programs that use only root-takings.

The authors analyze the computational complexity of sparse rational interpolation, and give the first deterministic algorithm for this problem with singly exponential bounds on the number of arithmetic operations.

The authors present the first algorithm for the (black box) interpolation of t-sparse, n-variate, rational functions without knowing bounds on exponents of their sparse representation, with the number of queries independent of exponents. In fact, the algorithm uses O(nt/sup t/) queries to the black box, and it can be implemented for a fixed t in a polynomially bounded storage (or polynomial parallel time).>

The authors consider the problem of reconstructing (i.e., interpolating) a t-sparse multivariate polynomial given a black box which will produce the value of the polynomial for any value of the arguments. It is shown that, if the polynomial has coefficients in a finite field $GF[q]$ and the black box can evaluate the polynomial in the field $GF[q^{\ulcorner 2\log_{q}(nt)+3 \urcorner}]$, where n is the number of variables, then there is an algorithm to interpolate the polynomial in $O(\log^3 (nt))$ boolean parallel time and $O(n^2 t^6 \log^2 nt)$ processors. This algorithm yields the first efficient deterministic polynomial time algorithm (and moreover boolean $NC$-algorithm) for interpolating t-sparse polynomials over finite fields and should be contrasted with the fact that efficient interpolation using a black box that only evaluates the polynomial at points in $GF[q]$ is not possible (cf. [M. Clausen, A. Dress, J. Grabmeier, and M. Karpinski, Theoret. Comput. Sci., 1990, to appear]). This algorithm, together with the efficient deterministic interpolation algorithms for fields of characteristic 0 (cf. [D. Yu. Grigoriev and M. Karpinski, in Proceedings of the 28th IEEE Symposium on the Foundations of Computer Science, 1987, pp. 166–172], [M. Ben-Or and P. Tiwari, in Proceedings of the 20th ACM Symposium on the Theory of Computing, 1988, pp. 301–309]), yields for the first time the general deterministic sparse conversion algorithm working over arbitrary fields. (The reason for this is that every field of positive characteristic contains a primitive subfield of this characteristic, and so this method can be applied to the slight extension of this subfield.) The method of solution involves the polynomial enumeration techniques of [D. Yu. Grigoriev and M. Karpinski, op. cit.] combined with introducing a new general method of solving the problem of determining if a t-sparse polynomial is identical to zero by evaluating it in a slight extension of the coefficient field (i.e., an extension whose degree over this field is logarithmic in nt).

In Part I of this paper, we give an extension of Liouville’s Theorem and give a number of examples which show that integration with special functions involves some phenomena that do not occur in integration with the elementary functions alone. Our main result generalizes Liouville’s Theorem by allowing, in addition to the elementary functions, special functions such as the error function, Fresnel integrals and the logarithmic integral (but not the dilogorithm or exponential integral) to appear in the integral of an elementary function. The basic conclusion is that these functions, if they appear, appear linearly. We give an algorithm which decides if an elementary function, built up using only exponential functions and rational operations has an integral which can be expressed in terms of elementary functions and error functions.

In this paper, we show that the theory of ordered differential fields has a model completion. We also show that any real differential field, finitely generated over the rational numbers, is isomorphic to some field of real meromorphic functions. In the last section of this paper, we combine these two results and discuss the problem of deciding if a system of differential equations has real analytic solutions. The author wishes to thank G. Stengle for some stimulating and helpful conversations and for drawing our attention to fields of real meromorphic functions. § 1. Real and ordered fields. A real field is a field in which −1 is not a sum of squares. An ordered field is a field F together with a binary relation < which totally orders F and satisfies the two properties: (1) If 0 < x and 0 < y then 0 < xy. (2) If x < y then, for all z in F, x + z < y + z. An element x of an ordered field is positive if x > 0. One can see that the square of any element is positive and that the sum of positive elements is positive. Since −1 is not positive, an ordered field is a real field. Conversely, given a real field F, it is known that one can define an ordering (not necessarily uniquely) on F [2, p. 274]. An ordered field F is a real closed field if: (1) every positive element is a square, and (2) every polynomial of odd degree with coefficients in F has a root in F. For example, the real numbers form a real closed field. Every ordered field can be embedded in a real closed field. It is also known that, in a real closed field K, polynomials satisfy the intermediate value property, i.e. if f(x) ∈ K[x] and a, b ∈ K, a < b, and f(a)f(b) < 0 then there is a c in K such that f(c) = 0.