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Multiplicative relations between the roots of a polynomial in Q [ x ] have drawn much attention in the field of arithmetic and algebra, while the problem of computing these relations is interesting to researchers in many other fields. In this paper, a sufficient condition is given for a polynomial f ∈ Q [ x ] to have only trivial multiplicative relations between its roots, which is a generalization of those sufficient conditions proposed in Smyth (1986) , Baron et al. (1995) and Dixon (1997). Based on the new condition, a subset E ⊂ Q [ x ] is defined and proved to be generic (i.e. , the set Q [ x ] \ E is very small). We develop an algorithm deciding whether a given polynomial f ∈ Q [ x ] is in E and returning a basis of the lattice consisting of the multiplicative relations between the roots of f whenever f ∈ E. The numerical experiments show that the new algorithm is very efficient for the polynomials in E. A large number of polynomials with much higher degrees, which were intractable before, can be handled successfully with the algorithm. [ABSTRACT FROM AUTHOR]

In this paper we describe the method which we applied to successfully compute the primary decomposition of a certain ideal coming from applications in combinatorial algebra and algebraic statistics regarding conditional independence statements with hidden variables. While our method is based on the algorithm for primary decomposition by Gianni, Trager and Zacharias, we were not able to decompose the ideal using the standard form of that algorithm, nor by any other method known to us. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm which, given a point and a set of polynomials, constructs a single cylindrical cell containing the point, such that the given polynomials are sign-invariant in the computed cell. To represent a single cylindrical cell, a novel data structure is introduced. The algorithm works with this representation and proceeds incrementally, i.e., first constructing a cell representing the whole real space, and then iterating over the input polynomials, refining the cell at each step to ensure the sign-invariance of the current input polynomial. A merge procedure realizing this refinement is described, and its correctness is proven. The merge procedure is based on McCallum's projection operator, but uses geometric information relative to the input point to reduce the number of projection polynomials computed. The use of McCallum's operator implies the incompleteness of the algorithm in general. However, the algorithm is complete for well-oriented sets of polynomials. Moreover, the incremental approach described in this paper can be easily adapted to a different projection operator. The cell construction algorithm presented in this paper is an alternative to the “model-based” method described by Jovanović and de Moura in a 2012 paper. It generalizes the algorithm described by the first author in a 2013 paper, which could only produce full-dimensional cells, to allow for the construction of cells of arbitrary dimension. While a thorough comparison, whether analytical or empirical, of the new algorithm and the “model-based” approach will be the subject of future work, this paper provides preliminary justification for asserting the superiority of the new method. [ABSTRACT FROM AUTHOR]

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]