Recovering Exact Results from Inexact Numerical Data in Algebraic Geometry

@article{Bates2013RecoveringER,
  title={Recovering Exact Results from Inexact Numerical Data in Algebraic Geometry},
  author={Daniel J. Bates and Jonathan D. Hauenstein and Timothy M. McCoy and Chris Peterson and Andrew J. Sommese},
  journal={Experimental Mathematics},
  year={2013},
  volume={22},
  pages={38 - 50}
}
Let be a set of homogeneous polynomials. Let Z denote the complex projective algebraic set determined by the zero locus of . Numerical-continuation-based methods can be used to produce arbitrary-precision numerical approximations of generic points on each irreducible component of Z. Consider the ideal and the prime decomposition over . This article illustrates how lattice-reduction algorithms may take as input numerically approximated generic points on Z and effectively extract exact elements… Expand
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References

SHOWING 1-10 OF 50 REFERENCES
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
TLDR
This article presents algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set of polynomial systems, by finding, at each dimension, generic points on each component. Expand
Numerical solution of multivariate polynomial systems by homotopy continuation methods
Let P ( x ) = 0 be a system of n polynomial equations in n unknowns. Denoting P = ( p 1 ,…, p n ), we want to find all isolated solutions of for x = ( x 1 ,…, x n ). This problem is very common inExpand
Symmetric Functions Applied to Decomposing Solution Sets of Polynomial Systems
TLDR
This paper proves theoretically and demonstrates in practice that linear traces suffice for this verification step, and shows how to do so more efficiently by building a structured grid of samples, using divided differences, and applying symmetric functions. Expand
NUMERICAL COMPUTATION OF THE DIMENSIONS OF THE COHOMOLOGY OF TWISTS OF IDEAL SHEAVES
This article presents several numerical algorithms for computations in sheaf cohomology. Let X be an algebraic set defined by a system of homogeneous multivariate polynomials with coefficients in C.Expand
A Numerical Local Dimension Test for Points on the Solution Set of a System of Polynomial Equations
TLDR
This article presents computational evidence to illustrate that the use of this new algorithm greatly reduces the cost of so-called “junk-point filtering,” previously a significant bottleneck in the computation of a numerical irreducible decomposition. Expand
SINGULAR: a computer algebra system for polynomial computations
TLDR
SINGULAR is a specialized computer algebra system for polynomial computations with emphasize on the needs of commutative algebra, algebraic geometry, and singularity theory, which features one of the fastest and most general implementations of various algorithms for computing standard resp. Expand
A numerical-symbolic algorithm for computing the multiplicity of a component of an algebraic set
TLDR
The main goal of this paper is to present an algorithm for determining the multiplicity of a component of an algebraic set that sidesteps the numerical stability issues which have obstructed other approaches by incorporating a combined numerical-symbolic technique. Expand
A course in computational algebraic number theory
  • H. Cohen
  • Computer Science, Mathematics
  • Graduate texts in mathematics
  • 1993
TLDR
The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods. Expand
The numerical solution of systems of polynomials - arising in engineering and science
Background: Polynomial Systems Homotopy Continuation Projective Spaces Probability One Polynomials of One Variable Other Methods Isolated Solutions: Coefficient-Parameter Homotopy PolynomialExpand
Direct methods for primary decomposition
SummaryLetI be an ideal in a polynomial ring over a perfect field. We given new methods for computing the equidimensional parts and radical ofI, for localizingI with respect to another ideal, andExpand
...
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3
4
5
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