Some Complexity Results for Polynomial Ideals

Mayr, Ernst

doi:10.1006/jcom.1997.0447

Cited by 46 publications

(31 citation statements)

References 51 publications

(57 reference statements)

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“…To our best knowledge, besides [18], the only known nontrivial complexity lower bounds for some of these quantities are in [2,62]. For other attempts to characterize the intrinsic complexity of problems of algebraic geometry, especially elimination, we refer to [36,50,51]. Capturing the complexity of some of the above problems will help to reduce the contrasts we mentioned at the beginning of this introduction.…”

Section: Introductionmentioning

confidence: 95%

Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets

Bürgisser

Cucker

2006

Journal of Complexity

102

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We define counting classes #P R and #P C in the Blum-Shub-Smale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over R, or of systems of polynomial equalities over C, respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over R) and algebraic sets (over C). We prove that the problem of computing the Euler-Yao characteristic of semialgebraic sets is FP #P R R -complete, and that the problem of computing the geometric degree of complex algebraic sets is FP #P C C -complete. We also define new counting complexity classes in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k ∈ N, the FPSPACE-hardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the Borel-Moore homology.

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Section: Introductionmentioning

confidence: 95%

Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets

Bürgisser

Cucker

2006

Journal of Complexity

102

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“…But, so far, algebraic methods have always been considered ''guilty'' of bad computational complexity, namely, the notorious bad complexity for computing general Gröbner bases when the number of variables grow (see [3] and references therein). This paper demonstrates that, by carefully analyzing the structure of toric ideals in particular problems, algebraic tools can compete (and win!)…”

Section: Introductionmentioning

confidence: 99%

Convex integer maximization via Graver bases

Loera

Hemmecke

Onn

et al. 2009

Journal of Pure and Applied Algebra

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MSC:We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R d and w 1 x, . . . , w d x are linear forms on R n ,}. This method works for arbitrary input data A, b, d, w 1 , . . . , w d , c. Moreover, for fixed d and several important classes of programs in variable dimension, we prove that our algorithm runs in polynomial time. As a consequence, we obtain polynomial time algorithms for various types of multi-way transportation problems, packing problems, and partitioning problems in variable dimension.

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“…An instance of HILBERT Z is a family of nonconstant homogeneous polynomials Remark 4.12. The algorithm in [6] combined with the upper bounds in [41] implies that HILBERT Z is in FEXPSPACE. We do not know of any better upper bound on this problem.…”

Section: Remark 47mentioning

confidence: 98%

The Complexity of Computing the Hilbert Polynomial of Smooth Equidimensional Complex Projective Varieties

Bürgisser

Lotz

2006

Found Comput Math

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We continue the study of counting complexity begun in [13], [14], [15] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be reduced in polynomial time to the problem of counting the number of complex common zeros of a finite set of multivariate polynomials. The reduction is based on a new formula for the coefficients of the Hilbert polynomial of a smooth variety. Moreover, we prove that the more general problem of computing the Hilbert polynomial of a homogeneous ideal is polynomial space hard. This implies polynomial space lower bounds for both the problems of computing the rank and the Euler characteristic of cohomology groups of coherent sheaves on projective space, improving the #P-lower bound in Bach [1].

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Some Complexity Results for Polynomial Ideals

Cited by 46 publications

References 51 publications

Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets

Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets

Convex integer maximization via Graver bases

The Complexity of Computing the Hilbert Polynomial of Smooth Equidimensional Complex Projective Varieties

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