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We develop the first algorithm for computing rational solutions of scalar integro-differential equations with polynomial coefficients. It starts by finding the possible poles of a rational solution. Then, bounding the order of each pole and solving an algebraic linear system, we compute the singular part of rational solutions at each possible pole. Finally, using partial fraction decomposition, the polynomial part of rational solutions is obtained by computing polynomial solutions of a non-homogeneous scalar integro-differential equation with a polynomial right-hand side. The paper is illustrated by examples where the computations are done with our Maple implementation. [ABSTRACT FROM AUTHOR]

This paper provides a new algorithm for the formal reduction of linear differential systems with Laurent series coefficients. We show how to obtain a decomposition of Balser, Jurkat and Lutz using eigenring techniques. This allows us to establish structural information on the obtained indecomposable subsystems and retrieve information on their invariants such as ramification. We show why classical algorithms then perform well on these subsystems. We also give precise estimates of the precision on the power series which is required in each step of our algorithm. The algorithm is implemented in Maple and examples are given in Saade (2018). [ABSTRACT FROM AUTHOR]

In this paper, a symbolic, algorithmic procedure to compute an immersion that recasts a polynomial system into a linear one up to an output injection is proposed. Such a technique is based on computing, through algebraic geometry methods, the set of all the embeddings of the system and on matching the coefficients of these polynomials with the ones of the embeddings of a linear system up to an output injection. The given algorithm is then relaxed to compute an immersion that recasts a polynomial system into a form that is linear up to a finite order and an output injection and to compute an approximation of the immersion. [ABSTRACT FROM AUTHOR]

This paper presents a new algorithm for implicitizing tensor product surfaces of bi-degree ( m , n ) with no base points, assuming that there are no moving planes of bi-degree ( m − 1 , n − 1 ) following the surface. The algorithm is based on some structural results: (1) There are exactly 2 n linearly independent moving planes of bi-degree ( m , n − 1 ) following the surface; (2) mn linearly independent moving quadrics of bi-degree ( m − 1 , n − 1 ) following the surface can be constructed from the 2 n linearly independent moving planes; (3) The mn linearly independent moving quadrics form a compact determinant of order mn which exactly gives the implicit equation of the rational surface. Complexity analysis and experimental results show that the new algorithm is significantly more efficient than the previous methods. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we summarize our recent work on establishing, for the first time, an algorithm for the symbolic solution of linear boundary problems. We put our work in the frame of Wen-Tsun Wu’s approach to algorithmic problem solving in analysis, geometry, and logic by mapping the significant aspects of the underlying domains into algebra. We briefly compare this with the lines of thought of Wolfgang Groebner. For building up the necessary tower of domains in a generic and flexible way, we use the machinery of algorithmic functors introduced in our Theorema project. The essence of this concept is explained in the first section of the paper. The main part of the paper then describes our symbolic analysis approach to linear boundary problems, which hinges on three basic principles: (1) Differentiation as well as integration is treated axiomatically, setting up an algebraic data structure that can encode the problem statement (differential equation and boundary conditions) and suitable symbolic expressions for their solution (Green’s operators qua integral operators). (2) Abstract boundary problems are introduced as pairs consisting of an epimorphism on a vector space (abstract differential operator) and a subspace of its dual (abstract boundary conditions). (3) Operator algebras are treated by noncommutative polynomials, modulo Groebner bases for certain relation ideals. [Copyright &y& Elsevier]

In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial n th Weyl algebra, the polynomial n th shift algebra, and Z n -graded polynomials in the n th q _ -Weyl algebra. The most unexpected result is that this noncommutative problem of factoring partial differential operators can be approached effectively by reducing it to the problem of solving systems of polynomial equations over a commutative ring. In the case where a given polynomial is Z n -graded, we can reduce the problem completely to factoring an element in a commutative multivariate polynomial ring. The implementation in Singular is effective on a broad range of polynomials and increases the ability of computer algebra systems to address this important problem. We compare the performance and output of our algorithm with other implementations in major computer algebra systems on nontrivial examples. [ABSTRACT FROM AUTHOR]

Abstract: The main goal of this paper is to present a new algorithm bounding the regularity and “alpha” (the lowest degree of existing hypersurface) of a linear system of hypersurfaces in passing through multiple points in a general position. To do the above we formulate and prove a new theorem, which allows us to show the non-speciality of a linear system by splitting it into non-special and simpler systems. As a result we give new bounds for multiple point Seshadri constants on . [Copyright &y& Elsevier]

Abstract: Latin squares can be seen as multiplication tables of quasigroups, which are, in general, non-commutative and non-associative algebraic structures. The number of Latin squares having a fixed isotopism in their autotopism group is at the moment an open problem. In this paper, we use Gröbner bases to describe an algorithm that allows one to obtain the previous number. Specifically, this algorithm is implemented in Singular to obtain the number of Latin squares related to any autotopism of Latin squares of order up to 7. [Copyright &y& Elsevier]

Abstract: A well-known result due to J.T. Stafford asserts that a stably free left module over the Weyl algebras or –where is a field of characteristic 0–with is free. The purpose of this paper is to present a new constructive proof of this result as well as an effective algorithm for the computation of bases of . This algorithm, based on the new constructive proofs [Hillebrand, A., Schmale, W., 2001. Towards an effective version of a theorem of Stafford. J. Symbolic Comput. 32, 699–716; Leykin, A., 2004. Algorithmic proofs of two theorems of Stafford. J. Symbolic Comput. 38, 1535–1550] of J.T. Stafford’s result on the number of generators of left ideals of , performs Gaussian elimination on the formal adjoint of the presentation matrix of . We show that J.T. Stafford’s result is a particular case of a more general one asserting that a stably free left -module with is free, where denotes the stable rank of a ring . This result is constructive if the stability of unimodular vectors with entries in can be tested. Finally, an algorithm which computes the left projective dimension of a general left -module defined by means of a finite free resolution is presented. It allows us to check whether or not the left -module is stably free. [Copyright &y& Elsevier]

Abstract: The main goal of this paper is to present an algorithm bounding the dimension of a linear system of plane curves of given degree (or monomial basis) with multiple points in general position. As a result we prove the Harbourne–Hirschowitz conjecture when the multiplicities of base points are bounded by 11. This gives a partial answer to the question of when bivariate polynomial interpolation is possible. [Copyright &y& Elsevier]

Abstract: In this paper, we present a new algorithm for the exact solutions of linear systems with integer coefficients using numerical methods. It terminates with the correct answer in well-conditioned cases or quickly aborts in ill-conditioned cases. Success of this algorithm on a linear equation requires that the linear system must be sufficiently well-conditioned for the numeric linear algebra method being used to compute a solution with sufficient accuracy. Our method is to find an initial approximate solution by using a numerical method, then amplify the approximate solution by a scalar, and adjust the amplified solution and corresponding residual to integers so that they can be computed without large integer arithmetic involved and can be stored exactly. Then we repeat these steps to refine the solution until sufficient accuracy is achieved, and finally reconstruct the rational solution. Our approximating, amplifying, and adjusting idea enables us to compute the solutions without involving high precision software floating point operations in the whole procedure or large integer arithmetic except at the final rational reconstruction step. We will expose the theoretical cost and show some experimental results. [Copyright &y& Elsevier]

Abstract: This paper is an informal introduction to differential Galois theory. It surveys recent work on differential Galois groups, related algorithms and some applications. [Copyright &y& Elsevier]