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Abstract The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized (max ⁡ , +) , Izhakian's extended and Izhakian–Rowen's supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. We show that in the case of idempotent semirings we can do this precisely when the Frobenius property is satisfied, that in the case of supertropical semirings this is always possible, and that in non-trivial symmetrized semirings this is never possible. Our results allow us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent. [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 describe software for an efficient factorization of polynomials over global fields . The algorithm for function fields was recently incorporated into our system KANT. The method is based on a generic algorithm developed by the second author in an earlier paper in this journal. Besides algorithmic aspects not contained in that paper we give details about the current implementation and about some complexity issues as well as a few illustrative examples. Also, a generalization of the application of LLL reduction for factoring polynomials over arbitrary global fields is developed. [Copyright &y& Elsevier]

The aim of this paper is to exploit the algorithms of paper Experimental Math. 8 (1999) in order to produce a new algebraic method for computing efficiently absolute Lagrange resolvent, a fundamental tool in constructive algebraic Galois theory. This article is composed of two parts.The main idea of the first part is to break up the calculation of absolute resolvent into smaller computations. Since a multi-resolvent is a factor of a resolvent, the whole resolvent may be computed by means of several multi-resolvents.The idea of the second part is that an irreducible polynomial over Z might be reducible over Z /pZ for certain integer p. So the first part can be applied and then, the Chinese remainder theorem allows to lift up the resolvent over Z. [Copyright &y& Elsevier]

In 1994 Lazard proposed an improved projection operation for cylindrical algebraic decomposition (CAD). For the proof he introduced a certain notion of valuation of a multivariate Puiseux series at a point. However a gap in one of the key supporting results for the improved projection was subsequently noticed. In this paper we show that Lazard's projection is valid for CAD construction for so-called well-oriented polynomial sets. Our proof does not make use of Lazard's notion of valuation, however. [ABSTRACT FROM AUTHOR]

Abstract: This paper describes an algorithm which produces all ideals compatible with a given surjective Frobenius near-splitting. [Copyright &y& Elsevier]

Abstract: This paper explores the certification of properties of an efficient functional program in the area of symbolic computation: the calculation of Gröbner basis of a set of multivariate polynomials. Our experience in the development of industrial systems and the certification of some of its relevant properties has led us to use a methodology consisting of functional programs to write the code and the formal verification of fundamental properties. The functional language objective caml has been chosen to implement the program. To verify the properties two approaches are explored: manual proofs that reason directly over the source code of the algorithms, applying techniques like equational reasoning, and theorem provers that are used as a tool to help us certify a model of the real system. The chosen proof assistant is coq. Not only will the certification of the software be taken into consideration but also its efficiency. In addition, we present a graphical interface which eases the use of the program. [Copyright &y& Elsevier]

Abstract: The Gröbner walk is an algorithm for conversion between Gröbner bases for different term orders. It is based on the polyhedral geometry of the Gröbner fan and involves tracking a line between cones representing the initial and target term order. An important parameter is the explicit numerical perturbation of this line. This usually involves both time and space, demanding arithmetic of integers much larger than the input numbers. In this paper we show how the explicit line may be replaced by a formal line using Robbiano’s characterization of group orders on . This gives rise to the generic Gröbner walk involving only Gröbner basis conversion over facets and computations with marked polynomials. The infinite precision integer arithmetic is replaced by term order comparisons between (small) integral vectors. This makes it possible to compute with infinitesimal numbers and perturbations in a consistent way without introducing unnecessary long integers. The proposed technique is closely related to the lexicographic (symbolic) perturbation method used in optimization and computational geometry. We report on an implementation of our algorithm specifically tailored to computations with lattice ideals. [Copyright &y& Elsevier]

Abstract: An algorithm is presented that computes explicit generators for the ring of differential operators on an orbifold, the quotient of a complex vector space by a finite group action. The algorithm also describes the relations among these generators. The algorithm presented in this paper is based on Schwarz’s study of a map carrying invariant operators to operators on the orbifold and on an algorithm to compute rings of invariants using Gröbner bases due to Derksen [Derksen, Harm, 1999. Computation of invariants for reductive groups. Adv. Math. 141 (2), 366–384]. It is also possible to avoid using Derksen’s algorithm, instead relying on the Reynolds operator and the Molien series. [Copyright &y& Elsevier]

Abstract: One of the main contributions which Volker Weispfenning made to mathematics is related to Gröbner bases theory. In this paper we present an algorithm for computing all algebraic intermediate subfields in a separably generated unirational field extension (which in particular includes the zero characteristic case). One of the main tools is Gröbner bases theory. Our algorithm also requires computing primitive elements and factoring over algebraic extensions. Moreover, the method can be extended to finitely generated -algebras. [Copyright &y& Elsevier]

Abstract: Consider the class of exp–log constants, which is constructed from the integers using the field operations, exponentiation and logarithm. Let be such an exp–log constant and let be its size as an expression. Witness conjectures attempt to give bounds for the number of decimal digits which need to be evaluated in order to test whether equals zero. For this purpose, it is convenient to assume that exponentials are only applied to arguments with absolute values bounded by 1. In that context, several witness conjectures have appeared in the literature and the strongest one states that it is possible to choose . In this paper we give a counterexample to this conjecture. We also extend it so as to cover similar, polynomial witness conjectures. [Copyright &y& Elsevier]

Abstract: Algebraic function fields of positive characteristic are non-perfect fields, and many standard algorithms for solving some fundamental problems in commutative algebra simply do not work over these fields. This paper presents practical algorithms for the first time for (1) computing the primary decomposition of ideals of polynomial rings defined over such fields and (2) factoring arbitrary multivariate polynomials over such fields. Difficulties involving inseparability and the situation where the transcendence degree is greater than one are completely overcome, while the algorithms avoid explicit construction of any extension of the input base field. As a corollary, the problem of computing the primary decomposition of a positive-dimensional ideal over a finite field is also solved. The algorithms perform very effectively in an implementation within the Magma Computer Algebra System, and an analysis of their practical performance is given. [Copyright &y& Elsevier]

Abstract: In this paper, we present different results related to bezoutian and residue theory. We consider, in particular, the problem of computing the structure of the quotient ring by an affine complete intersection, and an algorithm to obtain it, as conjectured in [Cardinal, J.-P., 1993. Dualité et algorithmes itératifs pour la résolution de systèmes polynomiaux. Ph.D. Thesis, Univ. de Rennes]. We analyze it in detail and prove the validity of the conjecture, for a modification of the initial method. Direct applications of the results in effective algebraic geometry are given. [Copyright &y& Elsevier]

The aim of this paper is to describe two new factorization algorithms for polynomials. The first factorizes polynomials modulo the prime ideal of a number field. The second factorizes polynomials over a number field. [Copyright &y& Elsevier]

In the computation of a Gröbner basis using Buchberger’s algorithm, a key issue for improving the efficiency is to produce techniques for avoiding as many unnecessary critical pairs as possible. A good solution would be to avoid all non-minimal critical pairs, and hence to process only a minimal set of generators of the module generated by the critical syzygies. In this paper we show how to obtain that desired solution in the homogeneous case while retaining the same efficiency as with the classical implementation. As a consequence, we get a new optimized Buchberger algorithm. [Copyright &y& Elsevier]

In this paper, we propose a semi-numerical algorithm for computing absolute factorization of multivariate polynomials. It is based on some properties appearing after a generic change of coordinate. Using numerical computation, Galois group action and rational approximation, this method provides an efficient probabilistic algorithm for medium degrees. Two implementations are presented and compared to other algorithms. [Copyright &y& Elsevier]

The short product of two power series is the meaningful part of the product of these objects, i.e., ∑i+j. Mulders (AAECC 11 (2000) 69) gives an algorithm to compute a short product faster than the full product in the case of Karatsuba’s multiplication (Karatsuba and Ofman, Dokl. Akad. Nauk SSSR 145 (1962) 293). This algorithm works by selecting a cutoff point k and performing a full k×k product and two (n−k)×(n−k) short products recursively. Mulders also gives a heuristically optimal cutoff point βn. In this paper, we determine the optimal cutoff point in Mulders’ algorithm. We also give a slightly more general description of Mulders’ method. [Copyright &y& Elsevier]

In this paper we prove that, for a given set of parametric primary surfaces and parametric clipping curves, all parametric blending solutions can be expressed as the addition of a particular parametric solution and a generic linear combination of the basis of a free module of rank 3. As a consequence, we present an algorithm that outputs a generic expression for all the parametric solutions for the blending problem. In addition, we also prove that the set of all polynomial parametric solutions (i.e. solutions that have polynomial parametrizations) for a parametric blending problem can also be expressed in terms of the basis of a free module of rank 3, and we prove an algorithmic criterion to decide whether there exist parametric polynomial solutions. As a consequence we also present an algorithm that decides the existence of polynomial solutions, and that outputs (if this type of solution exists) a generic expression for all polynomial parametric solutions for the problem. [Copyright &y& Elsevier]

The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper we refine parts of the algorithm to find Liouvillian solutions of third order equations. We show that, except for four finite groups and a reduction to the second order case, it is possible to give a formula in the imprimitive case. We also give necessary conditions and several simplifications for the computation of the minimal polynomial for the remaining finite set of finite groups (or any known finite group) by extracting ramification information from the character table. Several examples have been constructed, illustrating the possibilities and limitations. [Copyright &y& Elsevier]

Let An(k) be the Weyl algebra, with k a field of characteristic zero. It is known that every projective finitely generated left module is free or isomorphic to a left ideal. Let M be a left submodule of a free module. In this paper we give an algorithm to compute the projective dimension of M. If M is projective and rank(M)≥2 we give a procedure to find a basis. [Copyright &y& Elsevier]

In this paper an algorithm is given to decide if a given polynomial in two variables with coefficients in a finitely generated K-algebra is a coordinate and if so, find a mate for this polynomial. [Copyright &y& Elsevier]

In this paper, we propose algorithms for the following problems in the implicitization of a set of differential rational parametric equations P. (1) To find a characteristic set for the implicit prime ideal of P. (2) To find a canonical representation for the image of P. (3) To decide whether the parameters of P are independent, and if not, to re-parameterize P so that the new parametric equations have independent parameters. (4) To compute the inversion maps of P, and as a consequence, to decide whether P is proper. If the implicit variety is of dimension one and P is not proper, to find a proper re-parameterization for P. [Copyright &y& Elsevier]

In this paper we establish the Cayley expansion theory on factored and shortest expansions of typical Cayley expressions in two- and three-dimensional projective geometry. We set up a group of Cayley factorization formulae based on the classification of Cayley expansions. We continue to establish three powerful simplification techniques in bracket computation. On top of the Cayley expansions and simplifications, together with a set of elimination rules, we design an algorithm that can produce extremely short proofs in two- and three-dimensional projective geometry. The techniques developed here can be immediately applied to other symbolic computation tasks involving brackets. [Copyright &y& Elsevier]