Multilevel Polynomial Partitions and Simplified Range Searching

Matoušek, Jiřı́; Patáková, Zuzana

doi:10.1007/s00454-015-9701-2

Cited by 34 publications

(49 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These results have found applications in discrete geometry, for proving incidence bounds [11], as well as in effcient range-searching [20]. 1.…”

mentioning

confidence: 95%

“…As such they all depend on the maximum of the degrees of the polynomials used to define the given set or sign conditions. More recently, a new application of the bounds described above in discrete and computational geometry, triggered by the work of Guth and Katz [16], raised the question whether even the part of the bound in Theorem 1 that depends only on the degree d could have a finer dependence on the degrees of the polynomials in P and Q, in the case when the degrees of the polynomials in Q and those in P differ significantly (see [16,26,18,17,30,20]). This is one of the primary motivations behind the results proved in the current paper (see Section 1.2 below for more detail).…”

mentioning

confidence: 99%

See 1 more Smart Citation

On a real analog of Bezout inequality and the number of connected components of sign conditions

Barone

Basu

2016

Proc. London Math. Soc.

View full text Add to dashboard Cite

Let normalR be a real closed field and Q1,…,Qℓ∈R[X1,…,Xk] such that for each i,1⩽i⩽ℓ, deg(Qi)⩽di. For 1⩽i⩽ℓ, denote by MJX-tex-caligraphicscriptQi={Q1,…,Qi}, Vi the real variety defined by Qi, and ki an upper bound on the real dimension of Vi (by convention V0=Rk and k0=k). Suppose also that 2⩽d1⩽d2⩽1k+1d3⩽1(k+1)2d4⩽⋯⩽1(k+1)ℓ−3dℓ−1⩽1(k+1)ℓ−2dℓ, and that ℓ⩽k. We prove that the number of semi‐algebraically connected components of Vℓ is bounded by O(k)2k∏1⩽j<ℓdjkj−1−kjdℓkℓ−1. This bound can be seen as a weak extension of the classical Bezout inequality (which holds only over algebraically closed fields and is false over real closed fields) to varieties defined over real closed fields. Additionally, if P⊂R[X1,…,Xk] is a finite family of polynomials with deg(P)⩽d for all P∈P, cardP=s and dℓ⩽1k+1d, then we prove that the number of semi‐algebraically connected components of the realizations of all realizable sign conditions of the family MJX-tex-caligraphicscriptP restricted to Vℓ is bounded by O(k)2k(sd)kℓ∏1⩽j⩽ℓdjkj−1−kj. These results have found applications in discrete geometry, for proving incidence bounds [, ], as well as in efficient range searching [].

show abstract

“…These results have found applications in discrete geometry, for proving incidence bounds [11], as well as in effcient range-searching [20]. 1.…”

mentioning

confidence: 95%

mentioning

confidence: 99%

On a real analog of Bezout inequality and the number of connected components of sign conditions

Barone

Basu

2016

Proc. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…In the latter, we rely on the concept of polynomial partitioning (as introduced by Guth and Katz [20]) and combine it with a technique that relies on Hilbert polynomials. Recently, similar polynomial partitioning techniques were also studied by Matoušek and Safernová [32] and Basu and Sombra [6]. However, each of the three papers presents different proofs and very different results.…”

Section: Introductionmentioning

confidence: 99%

A semi-algebraic version of Zarankiewicz's problem

Fox

Pach

Sheffer³

et al. 2017

J. Eur. Math. Soc.

View full text Add to dashboard Cite

A bipartite graph G is semi-algebraic in R d if its vertices are represented by point sets P, Q ⊂ R d and its edges are defined as pairs of points (p, q) ∈ P × Q that satisfy a Boolean combination of a fixed number of polynomial equations and inequalities in 2d coordinates. We show that for fixed k, the maximum number of edges in a K k,kfree semi-algebraic bipartite graph G = (P, Q, E) in R 2 with |P | = m and |Q| = n is at most O((mn) 2/3 + m + n), and this bound is tight. In dimensions d ≥ 3, we show that all such semi-algebraic graphs have at most C (mn)where here ε is an arbitrarily small constant and C = C(d, k, t, ε). This result is a farreaching generalization of the classical Szemerédi-Trotter incidence theorem. The proof combines tools from several fields: VC-dimension and shatter functions, polynomial partitioning, and Hilbert polynomials.We also present various applications of our theorem. For example, a general pointvariety incidence bound in R d , an improved bound for a d-dimensional variant of the Erdős unit distances problem, and more.

show abstract

“…The following multi-level partitioning theorem is closely related to the partitioning theorem of Matoušek and Patáková [16]. However, the partitioning theorem in [16] is for points and hypersurfaces, while the present partitioning theorem exploits the fact that the varieties we wish to partition have the special intersection properties described in Lemma 5.8. This allows us to obtain a stronger result.…”

Section: Partitioning Points and Circles In Dual Spacementioning

confidence: 94%