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We describe differential rational normal forms of a rational function and their properties. Based on these normal forms, we present an algorithm which, given a hyperexponential function T(x), constructs two hyperexponential functions T;<sub>1;</sub>(x) and T;<sub>2;</sub>(x) such that T(x) = T;<sub>1;</sub><sup>'</sup>(x) + T;<sub>2;</sub>(x) and… (More)

This paper presents a modular algorithm for computing the greatest common right divisor (gcrd) of two univariate Ore polynomials over Z[t]. The subresultants of Ore polynomi-als are used to compute the evaluation homomorphic images of the gcrd. Rational number and rational function reconstructions are used to recover coefficients. The experimental results… (More)

When factoring linear partial differential systems with a finite-dimensional solution space or analyzing symmetries of nonlinear ode's, we need to look for rational solutions of certain nonlinear pde's. The nonlinear pde's are called Riccati-like because they arise in a similar way as Riccati ode's. In this paper we describe the structure of rational… (More)

We present an efficient algorithm for testing whether a given integral polynomial has two distinct roots a, B such that fflp is a root of unity. The test is based on results obtained by investigation of the structure of the splitting field of the polynomial. By this investigate ion, we found also an improved bound for the least common multiple of the orders… (More)

The subresultant theory for univariate commutative poly-nomials is generalized to Ore polynomials. The generalization includes: the subresultant theorem, gap structure, and subresultant algorithm. Using this generalization, we deene Sylvester's resultant o f t w o Ore polynomials, derive the respective determinantal formulas for the greatest common right… (More)

The long-term goal initiated in this work is to obtain fast algorithms and implementations for definite integration in Almkvist and Zeilberger's framework of (differential) creative telescoping. Our complexity-driven approach is to obtain tight degree bounds on the various expressions involved in the method. To make the problem more tractable, we restrict… (More)

The paper describes an algebraic construction of the inversive difference field associated with a discrete-time rational nonlinear control system under the assumption that the system is submersive. We prove that a system is submersive iff its associated difference ideal is proper, prime and reflexive. Next, we show that Kähler differentials of the… (More)

An orthogonal Ore ring is an abstraction of common properties of linear partial differential, shift and q-shift operators. Using orthogonal Ore rings, we present an algorithm for finding hyperexponential solutions of a system of linear differential, shift and q-shift operators, or any mixture thereof, whose solution space is finite-dimensional. The… (More)

Picard-Vessiot extensions for ordinary differential and difference equations are well known and are at the core of the associated Galois theories. In this paper, we construct fundamental matrices and Picard-Vessiot extensions for systems of linear partial functional equations having finite linear dimension. We then use those extensions to show that all the… (More)

The present paper applies the characteristic set method of algebraic differential polynomials to give a mechanical procedure which permits us to prove non-trivial theorems in the local theory of surfaces. By this method, we have discovered a new relation between the first and second fundamental forms of a surface in case this surface contains no umbilici. A… (More)