Degree bounds for Gröbner bases of low-dimensional polynomial ideals

Mayr, Ernst; Ritscher, Stephan

doi:10.1145/1837934.1837945

Cited by 9 publications

(12 citation statements)

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“…turns out to be a formidable challenge. The techniques in this paper in conjunction with the recent results in Gröbner basis theory [36], [37] show that an s.s.o.p. for K[V ] G can be constructed (or verified) in space that is exponential in n and time that is double exponential in n. This is so even requiring the cardinality of S to be only polynomial (instead of optimal) or subexponential.…”

Section: In Geometry and Complexity Theorymentioning

confidence: 57%

“…The upper bound on its running time in [36] depends only on the dimension of the variety, the dimension of the ambient space in which the variety is embedded, and a bound on the degrees of the generators of its ideal. The matching worst-case double exponential time and exponential space lower bound in [35], [37] for the running time of this Gröbner basis algorithm basically means that we can not go any further than EXPSPACE using such general algebraic geometry. What is needed is explicit algebraic geometry, or specifically, an explicit Gröbner basis (as formally defined in the full version) that depends on the deeper structure of the specific variety at hand.…”

Section: Historymentioning

confidence: 99%

“…Such a possibility cannot be ruled out at the outset, since the computation of the Gröbner basis, on which the current EXSPACE-algorithm for constructing an s.s.o.p. is based, is EXPSPACE-complete in general [35], [37].…”

Section: The Gct Chasmmentioning

confidence: 99%

“…Deterministic construction or even verification of ψ is however very difficult in general. For general V , the current best algorithms for deterministic construction and verification based on a recent fundamental advance [36] in Gröbner basis theory take in the worst case space that is exponential n and time that is double exponential in n. (Before [36] the time bound was double exponential in s, and hence, triple exponential in n.) Nothing better can be expected in general because [35], [37] also prove a matching exponential space lower bound for the computation of Gröbner basis in this general setting.…”

Section: Introductionmentioning

confidence: 99%

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Geometric Complexity Theory V: Equivalence between Blackbox Derandomization of Polynomial Identity Testing and Derandomization of Noether's Normalization Lemma

Mulmuley

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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It is shown that black-box derandomization of polynomial identity testing (PIT) is essentially equivalent to derandomization of Noether's Normalization Lemma for explicit algebraic varieties, the problem that lies at the heart of the foundational classification problem of algebraic geometry. Specifically:(1) It is shown that in characteristic zero black-box derandomization of the symbolic trace identity testing (STIT) brings the problem of derandomizing Noether's Normalization Lemma for the ring of invariants of the adjoint action of the general linear group on a tuple of matrices from EXPSPACE (where it is currently) to P. Next it is shown that assuming the Generalized Riemann Hypothesis (GRH), instead of the blackbox derandomization hypothesis, brings the problem from EXPSPACE to quasi-PH, instead of P. Thus black-box derandomization of STIT takes us farther than GRH. Variants of the main implication are also shown assuming, instead of the blackbox derandomization hypothesis in characteristic zero, Boolean lower bounds for constant-depth threshold circuits or uniform Boolean conjectures, in conjunction with GRH. These results may explain in a unified way why proving lower bounds or derandomization results for arithmetic circuits in characteristic zero or constant-depth Boolean threshold circuits, or proving uniform Boolean conjectures without relativizable proofs has turned out to be so hard, and also why GRH has turned out to be so hard from the complexity-theoretic perspective. Thus this investigation reveals that the foundational problems of Geometry (classification and GRH) and Complexity Theory (lower bounds and derandomization) share a common root difficulty that lies at the junction of these two fields. We refer to it as the GCT chasm.(2) It is shown that black-box derandomization of PIT in a strengthened form implies derandomization of Noether's Normalization Lemma in a strict form for any explicit algebraic variety.(3) Conversely, it is shown that derandomization of Noether's Normalization Lemma in a strict form for specific explicit varieties implies this strengthened form of blackbox derandomization of PIT and its various variants.(4) A unified geometric complexity theory (GCT) approach to derandomization and classification is formulated on the basis of this equivalence.(5) It is illustrated by showing that Noether's Normalization Lemma for the ring of invariants of any explicit linear action of a classical algebraic group can be quasi-derandomized (unconditionally) if the dimension of the group is constant.

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Section: In Geometry and Complexity Theorymentioning

confidence: 57%

Section: Historymentioning

confidence: 99%

Section: The Gct Chasmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Geometric Complexity Theory V: Equivalence between Blackbox Derandomization of Polynomial Identity Testing and Derandomization of Noether's Normalization Lemma

Mulmuley

2012

2012 IEEE 53rd Annual Symposium on Foundations of Computer Science

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“…The Jacobian matrix of the variety V = {x ∈ C n : u 1 (x) = · · · = The dimension of an ideal is related to properties of its Gröbner basis. For ideals of any dimension, the following bound on the degree of the Gröbner basis was shown in [48].…”

Section: Appendix A: Algebraic Geometrymentioning

confidence: 99%

An Improved Semidefinite Programming Hierarchy for Testing Entanglement

Harrow

Natarajan

2017

Commun. Math. Phys.

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We present a stronger version of the Doherty-Parrilo-Spedalieri (DPS) hierarchy of approximations for the set of separable states. Unlike DPS, our hierarchy converges exactly at a finite number of rounds for any fixed input dimension. This yields an algorithm for separability testing which is singly exponential in dimension and polylogarithmic in accuracy. Our analysis makes use of tools from algebraic geometry, but our algorithm is elementary and differs from DPS only by one simple additional collection of constraints.

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