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.     

Hitting Simplices with Points in ℝ3

The so-called first selection lemma states the following: given any set P of n points in ℝ ^d , there exists a point in ℝ ^d contained in at least c _d n ^d+1−O(n ^d ) simplices spanned by P, where the constant c _d depends on d. We present improved bounds on the first selection lemma in ℝ³. In particular, we prove that c ₃≥0.00227, improving the previous best result of c ₃≥0.00162 by Wagner (On k-sets and applications. Ph.D. thesis, ETH Zurich, 2003). This makes progress, for the three-dimensional case, on the open problems of Bukh et al. (Stabbing simplices by points and flats. Discrete Comput. Geom., 2010) (where it is proven that c ₃≤1/4⁴≈0.00390) and Boros and Füredi (The number of triangles covering the center of an n-set. Geom. Dedic. 17(1):69–77, 1984) (where the two-dimensional case was settled).     

Bichromatic and Equichromatic Lines in ℂ2 and ℝ2

Let G and R each be a finite set of green and red points, respectively, such that |G|=n, |R|=n−k, G∩R=∅, and the points of G∪R are not all collinear. Let t be the total number of lines determined by G∪R. The number of equichromatic lines (a subset of bichromatic) is at least (t+2n+3−k(k+1))/4. A slightly weaker lower bound exists for bichromatic lines determined by points in ℂ². For sufficiently large point sets, a proof of a conjecture by Kleitman and Pinchasi is provided. A lower bound of (2t+14n−k(3k+7))/14 is demonstrated for bichromatic lines passing through at most six points. Lower bounds are also established for equichromatic lines passing through at most four, five, or six points.     

The gonality of smooth curves with plane models

LetC be the normalization of an integral plane curve of degreed with δ ordinary nodes or cusps as its singularities. If δ=0, then Namba proved that there is no linear seriesg _d ^−2/1 and that everyg _d ^−1/1 is cut out by a pencil of lines passing through a point onC. The main purpose of this paper is to generalize his result to the case δ>0. A typical one is as follows: Ifd≥2(k+1), and δ<kd−(k+1)²+3 for somek>0, thenC has no linear seriesg _d ^−3/1 . We also show that ifd≥2k+3 and δ<kd−(k+1)²+2, then each linear seriesg _d ^−2/1 onC is cut out by a pencil of lines. We have similar results forg _d ^−1/1 andg _2d ^−9/1 . Furthermore, we also show that all of our theorems are sharp.     

Asymptotic Upper Bounds for Ramsey Functions

 We show that for any graph G with N vertices and average degree d, if the average degree of any neighborhood induced subgraph is at most a, then the independence number of G is at least Nf _{a +1}(d), where f _{a +1}(d)=∫₀ ¹(((1−t)^{1/( a +1)})/(a+1+(d−a−1)t))dt. Based on this result, we prove that for any fixed k and l, there holds r(K _k + _l ,K _n )≤ (l+o(1))n ^k /(logn) ^{k −1}. In particular, r(K _k , K _n )≤(1+o(1))n ^{k −1}/(log n) ^{k −2}. Received: May 11, 1998 Final version received: March 24, 1999     

Euclidean minimum spanning trees and bichromatic closest pairs

We present an algorithm to compute a Euclidean minimum spanning tree of a given setS ofN points inE ^d in timeO(F _d (N,N) log ^d N), whereF _d (n,m) is the time required to compute a bichromatic closest pair amongn red andm green points inE ^d . IfF _d (N,N)=Ω(N ^1+ε), for some fixed ɛ>0, then the running time improves toO(F _d (N,N)). Furthermore, we describe a randomized algorithm to compute a bichromatic closest pair in expected timeO((nm logn logm)^2/3+m log² n+n log² m) inE ₃, which yields anO(N ^4/3 log^4/3 N) expected time, algorithm for computing a Euclidean minimum spanning tree ofN points inE ³. Ind≥4 dimensions we obtain expected timeO((nm)^{1−1/([d/2]+1)+ε}+m logn+n logm) for the bichromatic closest pair problem andO(N ^{2−2/([d/2]+1)ε}) for the Euclidean minimum spanning tree problem, for any positive ɛ. The first, second, and fourth authors acknowledge support from the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS), a National Science Foundation Science and Technology Center under NSF Grant STC 88-09648. The second author's work was supported by the National Science Foundation under Grant CCR-8714565. The third author's work was supported by the Deutsche Forschungsgemeinschaft under Grant A1 253/1-3, Schwerpunktprogramm “Datenstrukturen und effiziente Algorithmen”. The last two authors' work was also partially supported by the ESPRIT II Basic Research Action of the EC under Contract No. 3075 (project ALCOM).     

Hypersurfaces in the Lorentz-Minkowski space satisfying Lkψ?=?Aψ?+?b

We study hypersurfaces in the Lorentz-Minkowski space \mathbbLⁿ⁺¹{\mathbb{L}^{n+1}} whose position vector ψ satisfies the condition L _k ψ = Aψ + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ? \mathbbR^(n+1)×(n+1){A\in\mathbb{R}^{(n+1)\times(n+1)}} is a constant matrix and b ? \mathbbLⁿ⁺¹{b\in\mathbb{L}^{n+1}} is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature, open pieces of totally umbilical hypersurfaces \mathbbSⁿ₁(r){\mathbb{S}^n_1(r)} or \mathbbHⁿ(-r){\mathbb{H}^n(-r)}, and open pieces of generalized cylinders \mathbbS^m₁(r)×\mathbbR^n-m{\mathbb{S}^m_1(r)\times\mathbb{R}^{n-m}}, \mathbbH^m(-r)×\mathbbR^n-m{\mathbb{H}^m(-r)\times\mathbb{R}^{n-m}}, with k + 1 ≤ m ≤ n − 1, or \mathbbL^m×\mathbbS^n-m(r){\mathbb{L}^m\times\mathbb{S}^{n-m}(r)}, with k + 1 ≤ n − m ≤ n − 1. This completely extends to the Lorentz-Minkowski space a previous classification for hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} given by Alías and Gürbüz (Geom. Dedicata 121:113–127, 2006).     

Equipartition of mass distributions by hyperplanes

We consider the problem of determining the smallest dimensiond=Δ(j, k) such that, for anyj mass distributions inR ^d , there arek hyperplanes so that each orthant contains a fraction 1/2 ^k of each of the masses. The case Δ(1,2)=2 is very well known. The casek=1 is answered by the ham-sandwich theorem with Δ(j, 1)=j. By using mass distributions on the moment curve the lower bound Δ(j, k)≥j(2 ^k −1)/k is obtained. We believe this is a tight bound. However, the only general upper bound that we know is Δ(j, k)≤j2 ^k−1. We are able to prove that Δ(j, k)=⌈j(^2k−1/k⌉ for a few pairs (j, k) ((j, 2) forj=3 andj=2 ⁿ withn≥0, and (2, 3)), and obtain some nontrivial bounds in other cases. As an intermediate result of independent interest we prove a Borsuk-Ulam-type theorem on a product of balls. The motivation for this work was to determine Δ(1, 4) (the only case forj=1 in which it is not known whether Δ(1,k)=k); unfortunately the approach fails to give an answer in this case (but we can show Δ(1, 4)≤5). This research was supported by the National Science Foundation under Grant CCR-9118874.     

On lattice points in polyhedral cross-sections

(a) We prove that the convex hull of anyk ^d +1 points of ad-dimensional lattice containsk+1 collinear lattice points. (b) For a convex polyhedron we consider the numbers of its lattice points in consecutive parallel lattice hyperplanes (levels). We prove that if a polyhedron spans no more than 2 ^d−1 levels, then this string of numbers may be arbitrary. On the other hand, we give an example of a string of 2 ^d−1+1 numbers to which no convex polyhedron corresponds inR ^d .     

Asymptotics of Recurrence Relation Coefficients,Hankel Determinant Ratios,and Root Products Associated with Laurent Polynomials Orthogonal with Respect to Varying Exponential Weights

Let Λ^ℝ denote the linear space over ℝ spanned by z ^k , k∈ℤ. Define the real inner product 〈⋅,⋅〉 _L :Λ^ℝ×Λ^ℝ→ℝ, , N∈ℕ, where V satisfies: (i) V is real analytic on ℝ∖{0}; (ii) lim  _{| x |→∞}(V(x)/ln (x ²+1))=+∞; and (iii) lim  _{| x |→0}(V(x)/ln (x ⁻²+1))=+∞. Orthogonalisation of the (ordered) base with respect to 〈⋅,⋅〉 _L yields the even degree and odd degree orthonormal Laurent polynomials (OLPs) : φ _2n (z)=∑ _k=−n ⁿ ξ _k ⁽²ⁿ⁾ z ^k , ξ _n ⁽²ⁿ⁾>0, and φ _2n+1(z)=∑ _k=−n−1 ⁿ ξ _k ⁽²ⁿ⁺¹⁾ z ^k , ξ _−n−1⁽²ⁿ⁺¹⁾>0. Associated with the even degree and odd degree OLPs are the following two pairs of recurrence relations: z φ _2n (z)=c _2n ^♯ φ _2n−2(z)+b _2n ^♯ φ _2n−1(z)+a _2n ^♯ φ _2n (z)+b _2n+1 ^♯ φ _2n+1(z)+c _2n+2 ^♯ φ _2n+2(z) and z φ _2n+1(z)=b _2n+1 ^♯ φ _2n (z)+a _2n+1 ^♯ φ _2n+1(z)+b _2n+2 ^♯ φ _2n+2(z), where c ₀ ^♯ =b ₀ ^♯ =0, and c _2k ^♯ >0, k∈ℕ, and z ⁻¹ φ _2n+1(z)=γ _2n+1 ^♯ φ _2n−1(z)+β _2n+1 ^♯ φ _2n (z)+α _2n+1 ^♯ φ _2n+1(z)+β _2n+2 ^♯ φ _2n+2(z)+γ _2n+3 ^♯ φ _2n+3(z) and z ⁻¹ φ _2n (z)=β _2n ^♯ φ _2n−1(z)+α _2n ^♯ φ _2n (z)+β _2n+1 ^♯ φ _2n+1(z), where β ₀ ^♯ =γ ₁ ^♯ =0, β ₁ ^♯ >0, and γ _2l+1 ^♯ >0, l∈ℕ. Asymptotics in the double-scaling limit N,n→∞ such that N/n=1+o(1) of the coefficients of these two pairs of recurrence relations, Hankel determinant ratios associated with the real-valued, bi-infinite strong moment sequence , and the products of the (real) roots of the OLPs are obtained by formulating the even degree and odd degree OLP problems as matrix Riemann-Hilbert problems on ℝ, and then extracting the large-n behaviours by applying the non-linear steepest-descent method introduced in (Ann. Math. 137(2):295–368, [1993]) and further developed in (Commun. Pure Appl. Math. 48(3):277–337, [1995]) and (Int. Math. Res. Not. 6:285–299, [1997]).      

Extensions of the D(?k2)-triples {k2, k2 ± 1, 4k2 ± 1}

Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a positive integer. In this paper, we show that if {k ², k ² + 1, 4k ² + 1, d} is a D(−k ²)-quadruple, then d = 1, and that if {k ² − 1, k ², 4k ² − 1, d} is a D(k ²)-quadruple, then d = 8k ²(2k ² − 1).     

A class of optimal linear codes of length one above the Griesmer bound

In this paper, we determine the smallest lengths of linear codes with some minimum distances. We construct a [g _q (k, d) + 1, k, d] _q code for sq ^k-1 − sq ^k-2 − q ^s  − q ² + 1 ≤ d ≤ sq ^k-1 − sq ^k-2 − q ^s with 3 ≤ s ≤ k − 2 and q ≥ s + 1. Then we get n _q (k, d) = g _q (k, d) + 1 for (k − 2)q ^k-1 − (k − 1)q ^k-2 − q ² + 1 ≤ d ≤ (k − 2)q ^k-1 − (k − 1)q ^k-2, k ≥ 6, q ≥ 2k − 3; and sq ^k-1 − sq ^k-2 − q ^s  − q + 1 ≤ d ≤ sq ^k-1 − sq ^k-2 − q ^s , s ≥ 2, k ≥ 2s + 1 and q ≥ 2s − 1. This work was partially supported by the Com²MaC-SRC/ERC program of MOST/KOSEF (grant ^# R11-1999-054) and was partially supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD)(KRF-2005-214-C00175).     

Bounding the number of connected components of a real algebraic set

For every polynomial mapf=(f ₁,…,f _k): ℝ ⁿ →ℝ ^k , we consider the number of connected components of its zero set,B(Z _f) and two natural “measures of the complexity off,” that is the triple(n, k, d), d being equal to max(degree off _i), and thek-tuple (Δ₁,...,Δ₄), Δ _k being the Newton polyhedron off _i respectively. Our aim is to boundB(Z _f) by recursive functions of these measures of complexity. In particular, with respect to (n, k, d) we shall improve the well-known Milnor-Thom’s bound μ _d (n)=d(2d−1) ⁿ⁻¹. Considered as a polynomial ind, μ _d (n) has leading coefficient equal to 2 ⁿ⁻¹. We obtain a bound depending onn, d, andk such that ifn is sufficiently larger thank, then it improves μ _d (n) for everyd. In particular, it is asymptotically equal to 1/2(k+1)n ^k−1 dⁿ, ifk is fixed andn tends to infinity. The two bounds are obtained by a similar technique involving a slight modification of Milnor-Thom's argument, Smith's theory, and information about the sum of Betti numbers of complex complete intersections.     

Refined enumerations of alternating sign matrices: monotone (<Emphasis Type="Italic">d</Emphasis>,<Emphasis Type="Italic">m</Emphasis>)-trapezoids with prescribed top and bottom row

Monotone triangles are plane integer arrays of triangular shape with certain monotonicity conditions along rows and diagonals. Their significance is mainly due to the fact that they correspond to n×n alternating sign matrices when prescribing (1,2,…,n) as bottom row of the array. We define monotone (d,m)-trapezoids as monotone triangles with m rows where the d−1 top rows are removed. (These objects are also equivalent to certain partial alternating sign matrices.) It is known that the number of monotone triangles with bottom row (k ₁,…,k _n ) is given by a polynomial α(n;k ₁,…,k _n ) in the k _i ’s. The main purpose of this paper is to show that the number of monotone (d,m)-trapezoids with prescribed top and bottom row appears as a coefficient in the expansion of a specialisation of α(n;k ₁,…,k _n ) with respect to a certain polynomial basis. This settles a generalisation of a recent conjecture of Romik et al. (Adv. Math. 222:2004–2035, 2009). Among other things, the result is used to express the number of monotone triangles with bottom row (1,2,…,i−1,i+1,…,j−1,j+1,…,n) (which is, by the standard bijection, also the number of n×n alternating sign matrices with given top two rows) in terms of the number of n×n alternating sign matrices with prescribed top and bottom row, and, by a formula of Stroganov for the latter numbers, to provide an explicit formula for the first numbers. (A formula of this type was first derived by Karklinsky and Romik using the relation of alternating sign matrices to the six-vertex model.)     

The primes contain arbitrarily long polynomial progressions

We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P ₁, …, P _k  ∈ Z[m] in one unknown m with P ₁(0) = … = P _k (0) = 0, and given any ε > 0, we show that there are infinitely many integers x and m, with 1 \leqslant m \leqslant x^e1 \leqslant m \leqslant x^\varepsilon, such that x + P ₁(m), …, x + P _k (m) are simultaneously prime. The arguments are based on those in [18], which treated the linear case P _j  = (j − 1)m and ε = 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties.     

Embeddings ofl p m intol ∞ n , 1≦p≦2

Isomorphic embeddings ofl _l ^m intol _∞ ⁿ are studied, and ford(n, k)=inf{‖T ‖ ‖T ⁻¹ ‖;T varies over all isomorphic embeddings ofl ₁ ^[klog2n] intol _∞ ⁿ we have that lim _n→∞ d(n, k)=γ(k)⁻¹,k>1, whereγ(k) is the solution of (1+γ)ln(1+γ)+(1 −γ)ln(1 −γ)=k ⁻¹ln4. Here [x] denotes the integer part of the real numberx.     

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     

On the representation of integers as sums of an odd number of squares

In a recent work, S. Cooper (J. Number Theory 103:135–162, [1988]) conjectured a formula for r _2k+1(p ²), the number of ways p ² can be expressed as a sum of 2k+1 squares. Inspired by this conjecture, we obtain an explicit formula for r _2k+1(n ²),n≥1. Dedicated to Srinivasa Ramanujan.     

On K 1,k -factorization of bipartite multigraphs

A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n, which partition the set of edges of λKm,n. In this paper, it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n, whenever k is any positive integer, is that （1） m ≤ kn, （2） n ≤ km, （3） km-n = kn-m ≡ 0 （mod （k^2- 1）） and （4） λ（km-n）（kn-m） ≡ 0 （mod k（k- 1）（k^2 - 1）（m ＋ n））.