A Survey on the Complexity of Solving Algebraic Systems

We present three algorithms in this paper: the first algorithm solves zero-dimensional parametric homogeneous polynomial systems with single exponential time in the number n of the unknowns, it decomposes the parameters space into a finite number of constructible sets and computes the finite number of solutions by parametric rational representations uniformly in each constructible set. The second algorithm factorizes absolutely multivariate parametric polynomials with single exponential time in n and in the degree upper bound d of the factorized polynomials. The third algorithm decomposes the algebraic varieties defined by parametric polynomial systems of positive dimension into absolutely irreducible components uniformly on the values of the parameters. The complexity bound of this algorithm is double-exponential in n. On the other hand, the complexity lower bound of the problem of resolution of parametric polynomial systems is double-exponential in n.

Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation - ISSAC '12, 2012

We study the complexity of computing the real solutions of a bivariate polynomial system using the recently proposed algorithm Bisolve [3]. Bisolve is a classical elimination method which first projects the solutions of a system onto the x-and y-axes and, then, selects the actual solutions from the so induced candidate set. However, unlike similar algorithms, Bisolve requires no genericity assumption on the input nor it needs any change of the coordinate system. Furthermore, extensive benchmarks from [3] confirm that the algorithm outperforms state of the art approaches by a large factor. In this work, we show that, for two polynomials f, g ∈ Z[x, y] of total degree at most n with integer coefficients bounded by 2 τ , Bisolve computes isolating boxes for all real solutions of the system f = g = 0 usingÕ(n 8 τ 2) bit operations 1 , thereby improving the previous record bound by a factor of at least n 2 .

Foundations of Computational Mathematics, 2003

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.

1995

We present a new method for solving symbolically zero-dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of an alternative data structure: arithmetic networks and straight-line programs with FOR gates. For sequential time complexity measured by the size of these networks we obtain the following result: 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. Here, the input is thought to be given by a straight-line program (or alternatively in sparse representation), and the length of the input is measured by number of variables, degree of equations and size of the program (or sparsity of the equations). Geometric degree has to be adequately defined. It is always bounded by the algebraic-combinatoric “Bézout number” of the system which is given by the Hilbert function of a suitable homogeneous ideal. However, in many important cases, the value of the geometric degree is much smaller than the Bézout number since it does not take into account multiplicities or degrees of extraneous components (which are at infinity in the affine case or contained in some coordinate hyperplane in the toric case). Finally, we announce the result that FOR gates can be avoided by a method which, based on Newton iteration, pulls back the whole question to ordinary arithmetic networks and straight-line programs. In this context, our complexity bounds remain valid. However, this second procedure is not rational anymore because it requires computing with algebraic numbers. This is due to its numeric ingredients (Newton iteration). Nevertheless, at least in the case of polynomial equation systems depending on parameters, the practical advantage of our method with respect to more traditional ones in symbolic and numeric computation is clearly visible. It should be well understood that our method does not improve the well known worst-case complexity bounds for zero-dimensional equation solving in symbolic and numeric computing.

Elimination theory is at the origin of algebraic geometry in the 19-th century and deals with 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 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 which are formulated and discussed both from the point of view of exact (i.e. symbolic) as well as from the point of view of approximative (i.e. numeric) computing. The mentioned results shall only be discussed informally. Proofs will appear elsewhere.

Journal of Symbolic Computation, 2009

This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of e OB(N 14 ) for the purely projection-based method, and e OB(N 12 ) for two subresultant-based methods: this notation ignores polylogarithmic factors, where N bounds the degree, and the bitsize of the polynomials. The previous record bound was e OB(N 14 ).

Journal of Symbolic Computation, 2007

We present a new algorithm for solving basic parametric constructible or semi-algebraic

2012

These pages contain a short overview on the state of the art of efficient numerical analysis methods that solve systems of multivariate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on the collaboration between Stephen Smale and Michael Shub, which set the foundations of this approach to polynomial system--solving, culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo, Peter Buergisser and Felipe Cucker.

Transactions of the American Mathematical Society, 1996

This paper deals with the description of the solutions of zero dimensional systems of polynomial equations. Based on different models for describing solutions, we consider suitable representations of a multiple root, or more precisely suitable descriptions of the primary component of the system at a root. We analyse the complexity of finding the representations and of algorithms which perform transformations between the different representations.

1997

This paper is devoted to the complexity analysis of a particular property, called "algebraic robustness" owned by all known symbolic methods of parametric polynomial equation solving (geometric elimination). It is shown that any parametric elimination procedure which owns this property must neccessarily have an exponential sequential time complexity.