A Gröbner Free Alternative for Polynomial System Solving

@article{Giusti2001AGF,
  title={A Gr{\"o}bner Free Alternative for Polynomial System Solving},
  author={Marc Giusti and Gr{\'e}goire Lecerf and Bruno Salvy},
  journal={J. Complex.},
  year={2001},
  volume={17},
  pages={154-211}
}
Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial, and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold… 

Tables from this paper

The Hardness of Polynomial Equation Solving
TLDR
This paper investigates the intrinsic sequential time complexity of universal elimination procedures for arbitrary continuous data structures encoding input and output objects of elimination theory and admitting the representation of certain limit objects.
On the Bit Complexity of Solving Bilinear Polynomial Systems
TLDR
A careful infinitesimal symbolic perturbation of the system allows us to treat degenerate and positive dimensional systems, thus making the algorithms and complexity analysis applicable to the general case.
Fast Algorithms for Zero-Dimensional Polynomial Systems using Duality
TLDR
This work addresses the question of speeding up the linear algebra phase for the computation of minimal polynomials and rational parametrizations in A using the A-module structure of the dual space $\widehat{A}$.
Algorithms for Discrete Differential Equations of Order 1
TLDR
It is proved that the total arithmetic size of the algebraic equations for F(t,1) is bounded polynomially in thesize of the input discrete differential equation, and that one can compute such equations in polynomial time.
EVALUATION TECHNIQUES FOR ZERO-DIMENSIONAL PRIMARY DECOMPOSITION EXTENDED ABSTRACT
In this talk, we will present an algorithm that computes the local algebra of the roots of a zero-dimensional polynomial equations system, whose cost is polynomial in the number of variables, in the
Algebraic Geometry Over Four Rings and the Frontier to Tractability
We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of
On the Topology of Real Algebraic Plane Curves
TLDR
This work revisits the problem of computing the topology and geometry of a real algebraic plane curve with a novelty of replacing Gröbner basis computations and isolation with rational univariate representations and induces a new approach for computing an arrangement of polylines isotopic to the input curve.
On the complexity of the F5 Gröbner basis algorithm
Solving Sparse Polynomial Systems using Gröbner Bases and Resultants
TLDR
This work will review these classical tools, their extensions, and recent progress in exploiting sparsity for solving polynomial systems, such as resultant computations, homotopy continuation methods, and most recently, Gröbner bases.
...
...

References

SHOWING 1-10 OF 190 REFERENCES
Lower bounds for diophantine approximations
When Polynomial Equation Systems Can Be "Solved" Fast?
TLDR
It is possible to solve any affine or toric zero-dimensional equation system in non-uniform sequential time which is polynomial in the length of the input description and the “geometric degree” of the equation system.
Solving Zero-Dimensional Systems Through the Rational Univariate Representation
  • F. Rouillier
  • Mathematics, Computer Science
    Applicable Algebra in Engineering, Communication and Computing
  • 1999
TLDR
It is shown that the solutions of any zero-dimensional system of polynomials can be expressed through a special kind of univariate representation (Rational Univariate Representation): where (f,g,g1, …,gn) are polynmials of K[X1,…, Xn].
Straight--Line Programs in Geometric Elimination Theory
Some algebraic and geometric computations in PSPACE
TLDR
A PSPACE algorithm for determining the signs of multivariate polynomials at the common zeros of a system of polynomial equations is given and it is shown that the existential theory of the real numbers can be decided in PSPACE.
La D Etermination Des Points Isol Es Et De La Dimension D'une Vari Et E Alg Ebrique Peut Se Faire En Temps Polynomial
We show that the dimension of an algebraic (a ne or projective) variety can be computed by a well parallelizable arithmetical network in non-uniform polynomial sequential time in the size of the
On the Time–Space Complexity of Geometric Elimination Procedures
TLDR
This paper applies the introduction of a new geometric invariant, called the degree of the input system, and the proof that the most common elimination problems have time complexity which is polynomial in this degree and the length of theinput.
The fundamental theorem of algebra and complexity theory
1. The main goal of this account is to show that a classical algorithm, Newton's method, with a standard modification, is a tractable method for finding a zero of a complex polynomial. Here, by
On the Complexity of Zero-dimensional Algebraic Systems
TLDR
A probabilistic algorithm is given which computes Grobner base for any ordering of its radical and/or all of its irreducible components in time d O(n) where d is the maximal degree of the polynomials and n the number of variables.
...
...