Bernstein-Sato ideals and hyperplane arrangements

We study the relation between zero loci of Bernstein-Sato ideals and roots of b-functions and obtain a criterion to guarantee that roots of b-functions of a reducible polynomial are determined by the zero locus of the associated Bernstein-Sato ideal. Applying the criterion together with a result of Maisonobe we prove that the set of roots of the b-function of a free hyperplane arrangement is determined by its intersection lattice.We also study the zero loci of Bernstein-Sato ideals and the associated relative characteristic cycles for arbitrary central hyperplane arrangements. We prove the multivariable $n/d$ conjecture of Budur for complete factorizations of arbitrary hyperplane arrangements, which in turn proves the strong monodromy conjecture for the associated multivariable topological zeta functions.     

From the Mahler Conjecture to Gauss Linking Integrals

We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product v(K)  = (Vol K)(Vol K°) of the volume of a symmetric convex body and its polar body K°. The Mahler conjecture asserts that the Mahler volume v(K) is minimized (non-uniquely) when K is an n-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domain is minimized when K is an ellipsoid. It implies the Mahler conjecture up to a factor of (π/4) ⁿ γ _n , where γ _n is a monotonic factor that begins at 4/π and converges to . This strengthens a result of Bourgain and Milman, who showed that there is a constant c such that the Mahler conjecture is true up to a factor of c ⁿ . The proof uses a version of the Gauss linking integral to obtain a constant lower bound on Vol K ^◇, with equality when K is an ellipsoid. It applies to a more general conjecture concerning the join of any two necks of the pseudospheres of an indefinite inner product space. Because the calculations are similar, we will also analyze traditional Gauss linking integrals in the sphere S ^n-1 and in hyperbolic space H ^n-1. Received: December 2006, Accepted: January 2007     

The relative isoperimetric inequality outside convex domains in R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We prove that the area of a hypersurface Σ which traps a given volume outside a convex domain C in Euclidean space R ⁿ is bigger than or equal to the area of a hemisphere which traps the same volume on one side of a hyperplane. Further, when C has smooth boundary ∂C, we show that equality holds if and only if Σ is a hemisphere which meets ∂C orthogonally.     

Pappus type theorems for hypersurfaces in a space form

In order to get further insight on the Weyl’s formula for the volume of a tubular hypersurface, we consider the following situation. Letc(t) be a curve in a space formM _λ ⁿ of sectional curvature λ. LetP ₀ be a totally geodesic hypersurface ofM _λ ⁿ throughc(0) and orthogonal toc(t). LetC ₀ be a hypersurface ofP ₀. LetC be the hypersurface ofM _λ ⁿ obtained by a motion ofC ₀ alongc(t). We shall denote it byC ^PorC ^Fif it is obtained by a parallel or Frenet motion, respectively. We get a formula for volume(C). Among other consequences of this formula we get that, ifc(0) is the centre of mass ofC ₀, then volume(C) ≥ volume(C),^P),and the equality holds whenC ₀ is contained in a geodesic sphere or the motion corresponds to a curve contained in a hyperplane of the Lie algebraO(n−1) (whenn=3, the only motion with these properties is the parallel motion). Work partially supported by a DGES Grant No. PB97-1425 and a AGIGV Grant No. GR0052.     

Coniveau 2 Complete Intersections and Effective Cones

Griffiths computation of the Hodge filtration on the cohomology of a smooth hypersurface X of degree d in \mathbbPⁿ{\mathbb{P}^n} shows that it has coniveau ≥ c once n ≥ dc. The generalized Hodge conjecture (GHC) predicts that the cohomology of X is then supported on a closed algebraic subset of codimension at least c. This is essentially unknown for c ≥ 2. In the case where c = 2, we exhibit a geometric phenomenon in the variety of lines of X explaining the estimate for the coniveau, and show that (GHC) would be implied in this case by the following conjecture on effective cones of cycles of intermediate dimension: Very moving subvarieties have their class in the interior of the effective cone.     

Better lower bounds on detecting affine and spherical degeneracies

We show that in the worst case, Ω(n ^d ) sidedness queries are required to determine whether a set ofn points in ℝ ^d is affinely degenerate, i.e., whether it containsd+1 points on a common hyperplane. This matches known upper bounds. We give a straightforward adversary argument, based on the explicit construction of a point set containing Ω(n ^d ) “collapsible” simplices, any one of which can be made degenerate without changing the orientation of any other simplex. As an immediate corollary, we have an Ω(n ^d ) lower bound on the number of sidedness queries required to determine the order type of a set ofn points in ℝ ^d . Using similar techniques, we also show that Ω(n ^d+1) in-sphere queries are required to decide the existence of spherical degeneracies in a set ofn points in ℝ ^d . An earlier version of this paper was presented at the 34th Annual IEEE Symposium on Foundations of Computer Science [8]. This research has been supported by NSF Presidential Young Investigator Grant CCR-9058440.     

Multivariate Regression Depth

   Abstract. The regression depth of a hyperplane with respect to a set of n points in \Real ^d is the minimum number of points the hyperplane must pass through in a rotation to vertical. We generalize hyperplane regression depth to k -flats for any k between 0 and d-1 . The k=0 case gives the classical notion of center points. We prove that for any k and d , deep k -flats exist, that is, for any set of n points there always exists a k -flat with depth at least a constant fraction of n . As a consequence, we derive a linear-time (1+ɛ) -approximation algorithm for the deepest flat. We also show how to compute the regression depth in time O(n ^d-2 +nlog n) when 1≤ k≤ d-2 .     

On the generalized critical values of a polynomial mapping

 Let be a polynomial dominant mapping and let deg f _i ≤d. We prove that the set K(f) of generalized critical values of f is contained in the algebraic hypersurface of degree at most D=(d+s(m−1)(d−1)) ⁿ , where . This implies in particular that the set B(f) of bifurcations points of f is contained in the hypersurface of degree at most D=(d+s(m−1)(d−1)) ⁿ . We give also an algorithm to compute the set K(f) effectively. Received: 11 June 2001 / Revised version: 1 July 2002 Published online: 24 January 2003 The author is partially supported by the KBN grant 2 PO3A 017 22. Mathematics Subject Classification (2000): 14D06, 14Q20, 51N10, 51N20, 15A04     

Depth in an Arrangement of Hyperplanes

A collection of n hyperplanes in ^d forms a hyperplane arrangement. The depth of a point is the smallest number of hyperplanes crossed by any ray emanating from θ . For d=2 we prove that there always exists a point θ with depth at least . For higher dimensions we conjecture that the maximal depth is at least . For arrangements in general position, an upper bound on the maximal depth is also established. Finally, we discuss algorithms to compute points with maximal depth. Received December 1, 1997, and in revised form June 6, 1998.     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter space S ^{n +1} ₁(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvature n(n-1)r is isometric to a sphere if r << c. Received: 18 December 1996 / Revised version: 26 November 1997     

Distribution of values of bounded generalized polynomials

A generalized polynomial is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication, and taking the integer part; a generalized polynomial mapping is a vector-valued mapping whose coordinates are generalized polynomials. We show that any bounded generalized polynomial mapping u: Z ^d  → R ^l has a representation u(n) = f(ϕ(n)x), n ∈ Z ^d , where f is a piecewise polynomial function on a compact nilmanifold X, x ∈ X, and ϕ is an ergodic Z ^d -action by translations on X. This fact is used to show that the sequence u(n), n ∈ Z ^d , is well distributed on a piecewise polynomial surface (with respect to the Borel measure on that is the image of the Lebesgue measure under the piecewise polynomial function defining ). As corollaries we also obtain a von Neumann-type ergodic theorem along generalized polynomials and a result on Diophantine approximations extending the work of van der Corput and of Furstenberg–Weiss.     

Estimates on Conjectures of Minkowski and woods

Let ℝ ⁿ be the n-dimensional Euclidean space. Let ∧ be a lattice of determinant 1 such that there is a sphere |X| < R which contains no point of ∧ other than the origin O and has n linearly independent points of ∧ on its boundary. A well known conjecture in the geometry of numbers asserts that any closed sphere in ℝ ⁿ of radius $ \sqrt {n/4} $ \sqrt {n/4} contains a point of ∧. This is known to be true for n ≤ 8. Here we give estimates on a more general conjecture of Woods for n ≥ 9. This leads to an improvement for 9 ≤ n ≤ 22 on estimates of Il’in (1991) to the long standing conjecture of Minkowski on product of n non-homogeneous linear forms.     

Stabbing Simplices by Points and Flats

The following result was proved by Bárány in 1982: For every d≥1, there exists c _d >0 such that for every n-point set S in ℝ ^d , there is a point p∈ℝ ^d contained in at least c _d n ^d+1−O(n ^d ) of the d-dimensional simplices spanned by S. We investigate the largest possible value of c _d . It was known that c _d ≤1/(2 ^d (d+1)!) (this estimate actually holds for every point set S). We construct sets showing that c _d ≤(d+1)^−(d+1), and we conjecture that this estimate is tight. The best known lower bound, due to Wagner, is c _d ≥γ _d :=(d ²+1)/((d+1)!(d+1) ^d+1); in his method, p can be chosen as any centerpoint of S. We construct n-point sets with a centerpoint that is contained in no more than γ _d n ^d+1+O(n ^d ) simplices spanned by S, thus showing that the approach using an arbitrary centerpoint cannot be further improved.     

Distributions of Points in <Emphasis Type="Italic">d</Emphasis> Dimensions and Large <Emphasis Type="Italic">k</Emphasis>-Point Simplices

We consider a variant of Heilbronn’s triangle problem by investigating for a fixed dimension d≥2 and for integers k≥2 with k≤d distributions of n points in the d-dimensional unit cube [0,1] ^d , such that the minimum volume of the simplices, which are determined by (k+1) of these n points is as large as possible. Denoting by Δ _k,d (n), the supremum of this minimum volume over all distributions of n points in [0,1] ^d , we show that c _k,d ⋅(log n)^1/(d−k+1)/n ^k/(d−k+1)≤Δ _k,d (n)≤c _k,d ′/n ^k/d for fixed 2≤k≤d, and, moreover, for odd integers k≥1, we show the upper bound Δ _k,d (n)≤c _k,d ″/n ^{k/d+(k−1)/(2d(d−1))}, where c _k,d ,c _k,d ′,c _k,d ″>0 are constants. A preliminary version of this paper appeared in COCOON ’05.     

The cardinality of the separated vertex set of a multidimensional cube

An n-dimensional cube and the sphere inscribed into it are considered. The conjecture of A. Ben-Tal, A. Nemirovski, and C. Roos states that each tangent hyperplane to the sphere strictly separates not more than 2 ⁿ⁻² cube vertices. In this paper this conjecture is proved for n ≤ 6. New examples of hyperplanes separating exactly 2 ⁿ⁻² cube vertices are constructed for any n. It is proved that hyperplanes orthogonal to radius vectors of cube vertices separate less than 2 ⁿ⁻² cube vertices for n ≥ 3.     

On the mixing time of a simple random walk on the super critical percolation cluster

 We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in Z ^d . We show that for d≥2 and p>p _c (Z ^d ), the mixing time of simple random walk on the largest cluster inside is Θ(n ² ) – thus the mixing time is robust up to a constant factor. The mixing time bound utilizes the Lovàsz-Kannan average conductance method. This is the first non-trivial application of this method which yields a tight result. Received: 16 December 2001 / Revised version: 13 August 2002 / Published online: 19 December 2002     

Generating Random Regular Graphs

Random regular graphs play a central role in combinatorics and theoretical computer science. In this paper, we analyze a simple algorithm introduced by Steger and Wormald [10] and prove that it produces an asymptotically uniform random regular graph in a polynomial time. Precisely, for fixed d and n with d = O(n^1/3−ε), it is shown that the algorithm generates an asymptotically uniform random d-regular graph on n vertices in time O(nd²). This confirms a conjecture of Wormald. The key ingredient in the proof is a recently developed concentration inequality by the second author. The algorithm works for relatively large d in practical (quadratic) time and can be used to derive many properties of uniform random regular graphs. * Research supported in part by grant RB091G-VU from UCSD, by NSF grant DMS-0200357 and by an A. Sloan fellowship.     

Oriented matroids with few mutations

This paper defines a “connected sum” operation on oriented matroids of the same rank. This construction is used for three different applications in rank 4. First it provides nonrealizable pseudoplane arrangements with a low number of simplicial regions. This contrasts the case of realizable hyperplane arrangements: by a classical theorem of Shannon every arrangement ofn projective planes in ℝP ^d-1 contains at leastn simplicial regions and every plane is adjacent to at leastd simplicial regions [17], [18]. We construct a class of uniform pseudoarrangements of 4n pseudoplanes in ℝP³ with only 3n+1 simplicial regions. Furthermore, we construct an arrangement of 20 pseudoplanes where one plane is not adjacent to any simplicial region. Finally we disprove the “strong-map conjecture” of Las Vergnas [1]. We describe an arrangement of 12 pseudoplanes containing two points that cannot be simultaneously contained in an extending hyperplane.     

Correlated equilibria of games with many players

Let G _m,n be the class of strategic games with n players, where each player has m≥2 pure strategies. We are interested in the structure of the set of correlated equilibria of games in G _m,n when n→∞. As the number of equilibrium constraints grows slower than the number of pure strategy profiles, it might be conjectured that the set of correlated equilibria becomes large. In this paper, we show that (1) the average relative measure of the set of correlated equilibria is smaller than 2⁻ⁿ; and (2) for each 1<c<m, the solution set contains c ⁿ correlated equilibria having disjoint supports with a probability going to 1 as n grows large. The proof of the second result hinges on the following inequality: Let c ₁, …, c _l be independent and symmetric random vectors in R ^k, l≥k. Then the probability that the convex hull of c ₁, …, c _l intersects R ^k ₊ is greater than or equal to . Received: December 1998/Final version: March 2000     

A Centrally Symmetric Version of the Cyclic Polytope

We define a centrally symmetric analogue of the cyclic polytope and study its facial structure. We conjecture that our polytopes provide asymptotically the largest number of faces in all dimensions among all centrally symmetric polytopes with n vertices of a given even dimension d=2k when d is fixed and n grows. For a fixed even dimension d=2k and an integer 1≤j<k we prove that the maximum possible number of j-dimensional faces of a centrally symmetric d-dimensional polytope with n vertices is at least for some c _j (d)>0 and at most as n grows. We show that c ₁(d)≥1−(d−1)⁻¹ and conjecture that the bound is best possible. Research of A. Barvinok partially supported by NSF grant DMS 0400617. Research of I. Novik partially supported by Alfred P. Sloan Research Fellowship and NSF grant DMS-0500748.