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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]

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]

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: 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]

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]

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]