Functional programming concepts and straight-line programs in computer algebra

Bruno, Nicolas; Heintz, Joos; Matera, Guillermo; Wachenchauzer, R.

doi:10.1016/s0378-4754(02)00035-6

Cited by 4 publications

(4 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Hardness of Polynomial Equation Solving 401 a robust (e.g., black-box) algorithm in the sense of Section 5.1 and that P can be applied to the encoding ω: A 1 ×A n → O n of the input object class of the flat family of zero-dimensional elimination problems (11). Then Theorem 3 implies that P requires exponential sequential time on infinitely many inputs.…”

Section: The Hardness Of Polynomial Equation Solvingmentioning

confidence: 99%

“…Therefore, from the point of view of "classical" parametric (i.e., branching-free) elimination procedures (based on the sparse or dense encoding of polynomials by their coefficients), it is not surprising that the sequential time becomes exponential in n for the computation of the solution of the general problem instance (11), even if we change the data structure representing the output objects (see, e.g., [30] and [56], [57] for this type of change of data structure).…”

Section: The Hardness Of Polynomial Equation Solvingmentioning

confidence: 99%

“…A successful implementation of the full algorithm ( [Lec00]) is based on [GLS01]. A partial implementation of the algorithm (including basic subroutines) is described in [BHMW02] (see also [CHLM00]). So far the account of successive improvements of data structures and algorithms for symbolic elimination.…”

mentioning

confidence: 99%

See 2 more Smart Citations

The Hardness of Polynomial Equation Solving

Castro

Giusti

Heintz

et al. 2003

Foundations of Computational Mathematics

Self Cite

View full text Add to dashboard Cite

Dedicated to Michel Demazure ... Il est fréquent, devant un problème concret, de trouver un théorème qui "s'applique presque".... Le rôle des contre-exemples est justement de délimiter le possible, et ce n'est pas par perversité (ou en tout cas pas totalement) que les textes mathématiques exhibent des monstres. M. DEMAZURE, 1987 Abstract. Elimination theory was at the origin of algebraic geometry in the nineteenth century and now deals with the algorithmic solving of multivariate polynomial equation systems over the complex numbers or, more generally, over an arbitrary algebraically closed field. In this paper we investigate the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory (i.e., polynomial equation systems) and admitting the representation of certain limit objects. Our main result is the following: let there be given such a data structure and together with this data structure a universal elimination algorithm, say P, solving arbitrary parametric polynomial equation systems. Suppose that the algorithm P avoids "unnecessary" branchings and that P admits the efficient computation of certain natural limit objects (as, e.g., the Zariski closure of a given constructible algebraic set or the parametric greatest common divisor of two given algebraic families of univariate polynomials). Then P cannot be a polynomial time algorithm.The paper contains different variants of this result and discusses their practical implications.

show abstract

Section: The Hardness Of Polynomial Equation Solvingmentioning

confidence: 99%

Section: The Hardness Of Polynomial Equation Solvingmentioning

confidence: 99%

See 1 more Smart Citation

The Hardness of Polynomial Equation Solving

Castro

Giusti

Heintz

et al. 2003

Foundations of Computational Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Straight-line programs, or equivalently algebraic circuits, are important both as a computational model and as a data structure for polynomial computation. Their rich history includes both algorithmic advances and practical implementations (Kaltofen, 1989;Sturtivant and Zhang, 1990;Bruno et al, 2002).…”

Section: Previous Workmentioning

confidence: 99%

Computer Algebra in Scientific Computing

England

Boulier

Sadykov

et al. 2013

Lecture Notes in Computer Science

View full text Add to dashboard Cite

We give a new probabilistic algorithm for interpolating a "sparse" polynomial f given by a straight-line program. Our algorithm constructs an approximation f * of f , such that f − f * probably has at most half the number of terms of f , then recurses on the difference f −f * . Our approach builds on previous work by Garg andSchost (2009), andRoche (2011), and is asymptotically more efficient in terms of the total cost of the probes required than previous methods, in many cases.

show abstract