Polar Varieties, Real Equation Solving and Data-Structures: The hypersurface case (original) (raw)

Polar Varieties and Efficient Real Equation Solving

1996

The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo [10] can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in [10] yields a method for symbolically solving a zerodimensional polynomial equation system in the affine (and toric) case. Its main feature is the use of adapted data structure: Arithmetical networks and straight-line programs. The algorithm solves any affine zerodimensional equation system in non-uniform sequential time that is polynomial in the length of the input description and an adequately defined affine degree of the equation system. Replacing the affine degree of the equation system by a suitably defined real degree of certain polar varieties associated to the input equation, which describes the hypersurface under consideration, and using straightline program codification of the input and intermediate results, we obtain a method for the problem introduced above that is polynomial in the input length and the real degree.

Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case

Journal of Complexity, 1996

The objective of this paper is to show how the recently proposed method by Giusti, Heintz, Morais, Morgenstern, Pardo \cite{gihemorpar} can be applied to a case of real polynomial equation solving. Our main result concerns the problem of finding one representative point for each connected component of a real bounded smooth hypersurface. The algorithm in \cite{gihemorpar} yields a method for

Polar varieties and efficient real elimination

Mathematische Zeitschrift, 2001

Let S 0 be a smooth and compact real variety given by a reduced regular sequence of polynomials f 1 , . . . , f p . This paper is devoted to the algorithmic problem of finding efficiently a representative point for each connected component of S 0 . For this purpose we exhibit explicit polynomial equations that describe the generic polar varieties of S 0 . This leads to a procedure which solves our algorithmic problem in time that is polynomial in the (extrinsic) description length of the input equations f 1 , . . . , f p and in a suitably introduced, intrinsic geometric parameter, called the degree of the real interpretation of the given equation system f 1 , . . . , f p .

On the complexity of real solving bivariate systems

Proceedings of the 2007 international symposium on Symbolic and algebraic computation - ISSAC '07, 2007

This paper is concerned with exact real solving of well-constrained, bivariate algebraic 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 O B (N 14 ) for the purely projectionbased method, and O B (N 12 ) for two subresultant-based methods: we ignore polylogarithmic factors, and N bounds the degree and the bitsize of the polynomials. The previous record bound was O B (N 16 ).

Bipolar varieties and real solving of a singular polynomial equation

2000

In this paper we introduce the concept of a bipolar variety of a real algebraic hypersurface. This notion is then used for the design and complexity estimations of a novel type of algorithms that finds algebraic sample point for the connected component of a singular real hypersurface. The complexity of these algorithms is polynomial in the maximal geometric degree of

Data Structures and Polynomial Equation Solving

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.

On the intrinsic complexity of point finding in real singular hypersurfaces

2009

In previous work we designed an efficient procedure that finds an algebraic sample point for each connected component of a smooth real complete intersection variety. This procedure exploits geometric properties of generic polar varieties and its complexity is intrinsic with respect to the problem. In the present paper we introduce a natural construction that allows to tackle the case of a non-smooth real hypersurface by means of a reduction to a smooth complete intersection.

Algorithms of intrinsic complexity for point searching in real singular hypersurfaces 1

2010

We treat the general problem of finding real solutions of multivariate polynomial equation systems in the case of a single equation F = 0 which is supposed to admit at least one F –regular real solution (where the gradient of F does not vanish) and which has possibly other, F –singular real solutions. We present two families of elimination algorithms of intrinsic complexity which solve this problem, one in the case that the real hypersurface defined by F is compact and another without this assumption. In worst case the complexity of our algorithms does not exceed the already known extrinsic complexity bound of (nd)O(n) for the elimination problem under consideration, where n is the number of indeterminates of F and d its (positive) degree. In the case Research partially supported by the following Argentinian, French and Spanish grants: CONICET PIP 2461/01, UBACYT X-098, PICT–2006–02067, BLAN NT05-4-45732 (projet GECKO), MTM 2007-62799. Humboldt-Universität zu Berlin, Institut für Ma...

Deformation Techniques for Efficient Polynomial Equation Solving

2000

Suppose we are given a parametric polynomial equation system encoded by an arithmetic circuit, which represents a generically flat and unramified family of zerodimensional algebraic varieties. Let us also assume that there is given the complete description of the solution of a particular unramified parameter instance of our system. We show that it is possible to``move'' the given particular solution along the parameter space in order to reconstruct by means of an arithmetic circuit the coordinates of the solutions of the system for an arbitrary parameter instance. The underlying algorithm is highly efficient, i.e., polynomial in the syntactic description of the input and the following geometric invariants: the number of solutions of a typical parameter instance and the degree of the polynomials occurring in the output. In fact, we prove a slightly more general result, which implies the previous statement by means of a well-known primitive element algorithm. We produce an efficient algorithmic description of the hypersurface obtained projecting polynomially the given generically flat family of varieties into a suitable affine space.

Algorithms of Intrinsic Complexity for Point Searching in Compact Real Singular Hypersurfaces

Foundations of Computational Mathematics, 2012

For a real squarefree multivariate polynomial F , we treat the general problem of finding real solutions of the equation F = 0 , provided that the real solution set {F = 0} R is compact. We admit that the equation F = 0 may have singular real solutions. We are going to decide whether this equation has a non-singular real solution and, if this is the case, we exibit one for each generically smooth connected component of {F = 0} R . We design a family of elimination algorithms of intrinsic complexity which solves this problem. In worst case the complexity of our algorithms does not exceed the already known extrinsic complexity bound of (nd) O(n) for the elimination problem under consideration, where n is the number of indeterminates of F and d its (positive) degree. In the case that the real variety defined by F is smooth, there exist already algorithms of intrinsic