## The complexity of computing the Hilbert polynomial of smooth equidimensional complex projective varieties (2005)

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Venue: | Foundations of Computational Mathematics |

Citations: | 3 - 2 self |

### BibTeX

@TECHREPORT{Bürgisser05thecomplexity,

author = {Peter Bürgisser and Martin Lotz},

title = {The complexity of computing the Hilbert polynomial of smooth equidimensional complex projective varieties},

institution = {Foundations of Computational Mathematics},

year = {2005}

}

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### Abstract

We continue the study of counting complexity begun in [11, 14, 13] 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. 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]. 1

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Citation Context ...5 Hilbert polynomial and degeneracy loci This section is devoted to the proof of Theorem 2.10. 5.1 Chern classes and Riemann-Roch References for the material presented here are [15, 27, 41]. See also =-=[21]-=- for the algebraic geometry perspective. Let V be a variety (recall the conventions made for varieties at the beginning of §2). Chern classes are characteristic cohomology classes ci(E) ∈ H 2i (V ) as... |

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Citation Context ...at these bounds are quite pessimistic and that for problems with “nice” geometry, single exponential upper bounds should hold for Gröbner bases. Among the results that are known in this direction are =-=[24, 18, 5, 38]-=-. However, currently no upper bound better than exponential space is known for the computation of the Hilbert function or Hilbert polynomial of a homogeneous ideal. Based on a lower bound on the homog... |

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Citation Context ...he embedding in projective space, the arithmetic genus is a birational invariant (cf. [26, Ex. III.5.3]). 4s2.2 Projective characters General references for the material presented in this section are =-=[20, 37]-=-. In the following we assume 0 ≤ m ≤ n. The Grassmann variety G(m,n) := {A | A ⊆ P n linear subspace of dimension m} is an irreducible smooth projective variety of dimension (m+1)(n−m) [25, Lect. 6]. ... |

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Citation Context ...t matchings in a bipartite graph, or equivalently, computing the permanent of its adjacency matrix, is #P-complete. For a comprehensive account to counting complexity we refer to [44, Chapter 18] and =-=[19]-=-. We now recall the definition of counting classes over C from [11, 13], which follows the lines used in discrete complexity theory to define #P and GapP [19]. We denote by � Z := Z ∪ {−∞, ∞,nil} the ... |

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Computation of Hilbert Functions
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Citation Context ...e variety V ⊆ P n . This polynomial encodes important information about the variety V , like its dimension, degree and arithmetic genus. Algorithms for computing Hilbert polynomials were described in =-=[42, 7, 6]-=-. Some of these algorithms have been implemented in computer algebra systems and work quite well in practice. These algorithms are based on the computation of Gröbner ∗ Institute of Mathematics, Unive... |

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Citation Context ...at these bounds are quite pessimistic and that for problems with “nice” geometry, single exponential upper bounds should hold for Gröbner bases. Among the results that are known in this direction are =-=[24, 18, 5, 38]-=-. However, currently no upper bound better than exponential space is known for the computation of the Hilbert function or Hilbert polynomial of a homogeneous ideal. Based on a lower bound on the homog... |

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Citation Context ...correspond to the special case λ = (1 k ) = (1,... ,1,0,... ,0), see [45, 10]. We remark that a different concept of generalized polar varieties has been previously used for algorithmic purposes, see =-=[2, 3]-=-. Note that the case where V is a linear space is degenerate: then dim ϕ(V ) = 0 and thus Pλ(F) is empty for almost all F ∈ F, provided |λ| > 0. A result by Zak, cf. [23, §7], states that this is the ... |

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Citation Context ...ert variety Ωλ(F) under the Gauss map. The wellknown polar varieties Pk(F) := P (1 k ) (F) = {x ∈ V | dim(TxV ∩ Fn−m+k−2) ≥ k − 1} correspond to the special case λ = (1 k ) = (1,... ,1,0,... ,0), see =-=[45, 10]-=-. We remark that a different concept of generalized polar varieties has been previously used for algorithmic purposes, see [2, 3]. Note that the case where V is a linear space is degenerate: then dim ... |

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Citation Context ...mputing the Hilbert polynomial of smooth equidimensional complex projective varieties Peter Bürgisser and Martin Lotz ∗ February 1, 2008 Abstract We continue the study of counting complexity begun in =-=[11, 14, 13]-=- 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 equidimensiona... |

18 |
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Citation Context ...eaf Z is in FEXPSPACE. 5 Hilbert polynomial and degeneracy loci This section is devoted to the proof of Theorem 2.10. 5.1 Chern classes and Riemann-Roch References for the material presented here are =-=[15, 27, 41]-=-. See also [21] for the algebraic geometry perspective. Let V be a variety (recall the conventions made for varieties at the beginning of §2). Chern classes are characteristic cohomology classes ci(E)... |

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Citation Context ...at these bounds are quite pessimistic and that for problems with “nice” geometry, single exponential upper bounds should hold for Gröbner bases. Among the results that are known in this direction are =-=[24, 18, 5, 38]-=-. However, currently no upper bound better than exponential space is known for the computation of the Hilbert function or Hilbert polynomial of a homogeneous ideal. Based on a lower bound on the homog... |

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Citation Context ...HNC in single exponential time (or even parallel polynomial time). A key point for showing this is the fact that a Gröbner basis of a zero-dimensional ideal can be computed in single exponential time =-=[18, 34, 35]-=-. The number of solutions can then be determined using linear algebra techniques, as described for example in [16, Chapter 2]. We remark that a corresponding counting class #P R over the reals has bee... |

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Topological Methods in Algebraic Geometry. Die Grundlehren der Mathematischen Wissenschaften, Band 131
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Citation Context ...g three steps: 1. We interpret the value pV (d) of the Hilbert polynomial of V ⊆ P n on d ∈ Z as the Euler characteristic χ(OV (d)) of the twisted sheaf OV (d). 2. The Hirzebruch-Riemann-Roch Theorem =-=[27]-=- gives an explicit combinatorial description of χ(OV (d)) in terms of certain determinants ∆λ(c) (related to Schur polynomials) in the Chern classes ci of the tangent bundle of V . 3. The homology cla... |

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Citation Context ...371 and Paderborn Institute for Scientific Computation (PaSCo). 1sbases, which leads to bad upper complexity estimates. In fact, the problem of computing a Gröbner basis is exponential space complete =-=[38]-=-. Both the cardinality and the maximal degree of a Gröbner basis might be doubly exponential in the number of variables [39, 28]. It is generally believed that these bounds are quite pessimistic and t... |

15 |
Computations of Hilbert-Poincaré Series
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Citation Context ...e variety V ⊆ P n . This polynomial encodes important information about the variety V , like its dimension, degree and arithmetic genus. Algorithms for computing Hilbert polynomials were described in =-=[42, 7, 6]-=-. Some of these algorithms have been implemented in computer algebra systems and work quite well in practice. These algorithms are based on the computation of Gröbner ∗ Institute of Mathematics, Unive... |

14 |
Young tableaux, volume 35 of London Mathematical Society Student Texts
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Citation Context ...he embedding in projective space, the arithmetic genus is a birational invariant (cf. [26, Ex. III.5.3]). 4s2.2 Projective characters General references for the material presented in this section are =-=[20, 37]-=-. In the following we assume 0 ≤ m ≤ n. The Grassmann variety G(m,n) := {A | A ⊆ P n linear subspace of dimension m} is an irreducible smooth projective variety of dimension (m+1)(n−m) [25, Lect. 6]. ... |

13 | The determinantal formula of Schubert calculus - Kempf, Laksov - 1974 |

13 |
A single exponential bound on the complexity of computing Gröbner bases of zero-dimensional ideals
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Citation Context ...HNC in single exponential time (or even parallel polynomial time). A key point for showing this is the fact that a Gröbner basis of a zero-dimensional ideal can be computed in single exponential time =-=[18, 34, 35]-=-. The number of solutions can then be determined using linear algebra techniques, as described for example in [16, Chapter 2]. We remark that a corresponding counting class #P R over the reals has bee... |

12 | eds) Some Tapas of Computer Algebra - Cohen, Cuypers, et al. - 1999 |

11 | Sheaf algorithms using the exterior algebra
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Citation Context ...ombined with the upper bounds in [38] implies that Hilbert Z is in FEXPSPACE. We do not know of any better upper bound on this problem. The known algorithms for sheaf cohomology (cf. [50, Chapter 8], =-=[17]-=-) suggest that RankSheaf Z is in FEXPSPACE. 5 Hilbert polynomial and degeneracy loci This section is devoted to the proof of Theorem 2.10. 5.1 Chern classes and Riemann-Roch References for the materia... |

10 | Variations by complexity theorists on three themes of Euler, Bézout, Betti, and Poincaré - Bürgisser, Cucker - 2004 |

10 | The real dimension problem is NPR-complete
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Citation Context ...ch that the projection of Su on this subspace has a nonempty interior. Writing this condition as a first order formula over R yields the claim for the dimension. (For a more economic description, see =-=[33]-=-.) Let Bǫ(x) denote the open ball with radius ǫ centered at x. We have dimx Su ≥ d if and only if dim(Su ∩ Bǫ(x)) ≥ d for sufficiently small ǫ > 0, cf. [4]. Writing this as a first order formula over ... |

8 |
Counting problems over the reals
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Citation Context ...ions can then be determined using linear algebra techniques, as described for example in [16, Chapter 2]. We remark that a corresponding counting class #P R over the reals has been introduced by Meer =-=[40]-=- and was further explored in [11]. 3.2 Polynomial hierarchy over the reals The constant-free polynomial hierarchy over the reals will be needed in the next section for extending the notion of a parsim... |

8 |
Sulle intersezioni delle varieta algebriche e sopra i loro caratteri e singolarita proiettive, Torino Mem
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Citation Context ...) = |λ|, (iii) there exists an integer dλ, such that deg Pλ(F) = dλ, provided ϕ ⋔ Ωλ(F). We call deg Pλ := dλ the projective character of V corresponding to λ. These quantities were studied by Severi =-=[46]-=-, see also [21, Ex. 14.3.3]. Note that the degree of V equals the projective character for λ = 0. Example 2.7 Let V ⊆ P 2 be a smooth curve. Then deg P1 counts the number of points on the curve whose ... |

7 |
Algebraic Geometry. GTM
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Citation Context ...ogy groups of coherent sheaves on projective space. The lower bound is also true for the problem to compute the corresponding Euler characteristic. For an introduction to sheaf cohomology we refer to =-=[26, 29]-=-. We encode the input to our problems as in [1]. Thus we specify a coherent sheaf on Pn by giving a graded matrix. This is a matrix (pij)1≤i≤s,1≤j≤r of homogeneous polynomials in S := C[X0,... ,Xn] to... |

5 |
Algebraic geometry, volume 76 of Graduate Texts in Mathematics
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Citation Context ...ogy groups of coherent sheaves on projective space. The lower bound is also true for the problem to compute the corresponding Euler characteristic. For an introduction to sheaf cohomology we refer to =-=[26, 29]-=-. We encode the input to our problems as in [1]. Thus we specify a coherent sheaf on Pn by giving a graded matrix. This is a matrix (pij)1≤i≤s,1≤j≤r of homogeneous polynomials in S := C[X0,... ,Xn] to... |

5 | Topology and Geometry. Number 139 - Bredon - 1993 |

4 |
The computation of the Hilbert function
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(Show Context)
Citation Context ...e variety V ⊆ P n . This polynomial encodes important information about the variety V , like its dimension, degree and arithmetic genus. Algorithms for computing Hilbert polynomials were described in =-=[42, 7, 6]-=-. Some of these algorithms have been implemented in computer algebra systems and work quite well in practice. These algorithms are based on the computation of Gröbner ∗ Institute of Mathematics, Unive... |

3 | Representation Theory. Number 129 - Fulton, Harris - 1991 |

2 |
Sheaf cohomology is #P-hard
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Citation Context ...lynomial 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]-=-. 1 Introduction Despite the impressive progress in the development of algebraic algorithms and computer algebra packages, the inherent computational complexity of even the most basic problems in alge... |

2 |
classes to Milnor classes—a history of characteristic classes for singular varieties
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(Show Context)
Citation Context ...ert variety Ωλ(F) under the Gauss map. The wellknown polar varieties Pk(F) := P (1 k ) (F) = {x ∈ V | dim(TxV ∩ Fn−m+k−2) ≥ k − 1} correspond to the special case λ = (1 k ) = (1,... ,1,0,... ,0), see =-=[45, 10]-=-. We remark that a different concept of generalized polar varieties has been previously used for algorithmic purposes, see [2, 3]. Note that the case where V is a linear space is degenerate: then dim ... |

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1 |
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Citation Context ...uerg@upb.de [Extended Abstract] ∗ Institute of Mathematics University of Paderborn D-33095 Paderborn, Germany Martin Lotz † lotzm@upb.de ABSTRACT We continue the study of counting complexity begun in =-=[7, 8, 9]-=- 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 equidimensiona... |