2,868 research outputs found
Analysis of Randomized Algorithms in Real Algebraic Geometry
Consider the problem of computing at least one point in each connected component of a smooth real algebraic set. This is a basic and important operation in real and semi-algebraic geometry: it gives an upper bound on the number of connected components of the algebraic set, it can be used to decide if the algebraic set has real solutions, and it is also used as a subroutine in many higher-level algorithms.
We consider an algorithm for this problem by Safey El Din and Schost: "Polar varieties and computation of one point in each connected component of a smooth real algebraic set," (ISSAC'03). This algorithm uses random changes of variables that are proven to generically ensure certain desirable geometric properties. The cost of the algorithm was given in an algebraic complexity model, and the analysis of the bit complexity and the error probability were left for future work.
We also consider another algorithm that solves a special case of the problem. Namely, when the algebraic set is a compact hypersurface.
We determine the bit complexity and error probability of these algorithms. Our main contribution is a quantitative analysis of several genericity statements, such as Thom's weak transversality theorem and Noether normalization properties for polar varieties. Furthermore, in doing this work, we have developed techniques that can be used in the analysis of further randomized algorithms in real algebraic geometry, which rely on related genericity properties.
Polar Varieties and Efficient Real Elimination
Let be a smooth and compact real variety given by a reduced regular
sequence of polynomials . This paper is devoted to the
algorithmic problem of finding {\em efficiently} a representative point for
each connected component of . For this purpose we exhibit explicit
polynomial equations that describe the generic polar varieties of . This
leads to a procedure which solves our algorithmic problem in time that is
polynomial in the (extrinsic) description length of the input equations and in a suitably introduced, intrinsic geometric parameter, called
the {\em degree} of the real interpretation of the given equation system .Comment: 32 page
Polar Varieties, Real Equation Solving and Data-Structures: The hypersurface case
In this paper we apply for the first time a new method for multivariate
equation solving which was developed in \cite{gh1}, \cite{gh2}, \cite{gh3} for
complex root determination to the {\em real} case. Our main result concerns the
problem of finding at least one representative point for each connected
component of a real compact and smooth hypersurface. The basic algorithm of
\cite{gh1}, \cite{gh2}, \cite{gh3} yields a new method for symbolically solving
zero-dimensional polynomial equation systems over the complex numbers. One
feature of central importance of this algorithm is the use of a
problem--adapted data type represented by the data structures arithmetic
network and straight-line program (arithmetic circuit). The algorithm finds the
complex solutions of any affine zero-dimensional equation system in non-uniform
sequential time that is {\em polynomial} in the length of the input (given in
straight--line program representation) and an adequately defined {\em geometric
degree of the equation system}. Replacing the notion of geometric degree of the
given polynomial equation system by a suitably defined {\em real (or complex)
degree} of certain polar varieties associated to the input equation of the real
hypersurface under consideration, we are able to find for each connected
component of the hypersurface a representative point (this point will be given
in a suitable encoding). The input equation is supposed to be given by a
straight-line program and the (sequential time) complexity of the algorithm is
polynomial in the input length and the degree of the polar varieties mentioned
above.Comment: Late
A Special Homotopy Continuation Method For A Class of Polynomial Systems
A special homotopy continuation method, as a combination of the polyhedral
homotopy and the linear product homotopy, is proposed for computing all the
isolated solutions to a special class of polynomial systems. The root number
bound of this method is between the total degree bound and the mixed volume
bound and can be easily computed. The new algorithm has been implemented as a
program called LPH using C++. Our experiments show its efficiency compared to
the polyhedral or other homotopies on such systems. As an application, the
algorithm can be used to find witness points on each connected component of a
real variety
Polar Varieties and Efficient Real Equation Solving: The Hypersurface Case
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 symbolically solving a zero-dimensional 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 zero-dimensional equation system in non-uniform sequential
time that is polynomial in the length of the input description and an
adequately defined {\em affine degree} of the equation system. Replacing the
affine degree of the equation system by a suitably defined {\em real degree} of
certain polar varieties associated to the input equation, which describes the
hypersurface under consideration, and using straight-line 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.Comment: Late
Nearest Points on Toric Varieties
We determine the Euclidean distance degree of a projective toric variety.
This extends the formula of Matsui and Takeuchi for the degree of the
-discriminant in terms of Euler obstructions. Our primary goal is the
development of reliable algorithmic tools for computing the points on a real
toric variety that are closest to a given data point.Comment: 20 page
The bottleneck degree of algebraic varieties
A bottleneck of a smooth algebraic variety is a pair
of distinct points such that the Euclidean normal spaces at
and contain the line spanned by and . The narrowness of bottlenecks
is a fundamental complexity measure in the algebraic geometry of data. In this
paper we study the number of bottlenecks of affine and projective varieties,
which we call the bottleneck degree. The bottleneck degree is a measure of the
complexity of computing all bottlenecks of an algebraic variety, using for
example numerical homotopy methods. We show that the bottleneck degree is a
function of classical invariants such as Chern classes and polar classes. We
give the formula explicitly in low dimension and provide an algorithm to
compute it in the general case.Comment: Major revision. New introduction. Added some new illustrative lemmas
and figures. Added pseudocode for the algorithm to compute bottleneck degree.
Fixed some typo
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