Van Der Waerden Conjecture Research Articles

Multidimensional permanents of polystochastic matrices

A d-dimensional matrix is called 1-polystochastic if it is non-negative and the sum over each line equals 1. Such a matrix that has a single 1 in each line and zeros elsewhere is called a 1-permutation matrix. A diagonal of a d-dimensional matrix of order n is a choice of n elements, no two in the same hyperplane. The permanent of a d-dimensional matrix is the sum over the diagonals of the product of the elements within the diagonal.For a given order n and dimension d, the set of 1-polystochastic matrices forms a convex polytope that includes the 1-permutation matrices within its set of vertices. For even n and odd d, we give a construction for a class of 1-permutation matrices with zero permanent. Consequently, we show that the set of 1-polystochastic matrices with zero permanent contains at least nn3/2(1/2−o(1)) 1-permutation matrices and contains a polytope of dimension at least cn3/2 for fixed c,d and even n→∞. We also provide counterexamples to a conjecture by Taranenko [11] about the location of local extrema of the permanent.For odd d, we give a construction of 1-permutation matrices that decompose into a convex linear sum of positive diagonals. These combine with a theorem of Taranenko [11] to provide counterexamples to a conjecture by Dow and Gibson [4] generalising van der Waerden's conjecture to higher dimensions.

The scaling mean and a Law of Large Permanents

In this paper we study two types of means of the entries of a nonnegative matrix: the permanental mean, which is defined using permanents, and the scaling mean, which is defined in terms of an optimization problem. We explore relations between these two means, making use of important results by Egorychev and Falikman (the van der Waerden conjecture), Friedland, Sinkhorn, and others. We also define a scaling mean for functions in a much more general context. Our main result is a Law of Large Permanents, a pointwise ergodic theorem for permanental means of dynamically defined matrices that expresses the limit as a functional scaling mean. The concepts introduced in this paper are general enough so to include as particular cases certain classical types of means, as for example symmetric means and Muirhead means. As a corollary, we reobtain a formula of Halász and Székely for the limit of the symmetric means of a stationary random process.

A note on extrema of linear combinations of elementary symmetric functions

This note provides a new approach to a result of Foregger [T.H. Foregger, On the relative extrema of a linear combination of elementary symmetric functions, Linear Multilinear Algebra 20 (1987) pp. 377–385] and related earlier results by Keilson [J. Keilson, A theorem on optimum allocation for a class of symmetric multilinear return functions, J. Math. Anal. Appl. 15 (1966), pp. 269–272] and Eberlein [P.J. Eberlein, Remarks on the van der Waerden conjecture, II, Linear Algebra Appl. 2 (1969), pp. 311–320]. Using quite different techniques, we prove a more general result from which the others follow easily. Finally, we argue that the proof in [Foregger, 1987] is flawed.

Van der Waerden/Schrijver-Valiant like Conjectures and Stable (aka Hyperbolic) Homogeneous Polynomials: One Theorem for all

Let $p$ be a homogeneous polynomial of degree $n$ in $n$ variables, $p(z_1,...,z_n) = p(Z) , Z \in C^{n}$. We call a such polynomial $p$ H-Stable if $p(z_1,...,z_n) \neq 0$ provided the real parts $Re(z_i) > 0, 1 \leq i \leq n$. This notion from Control Theory is closely related to the notion of Hyperbolicity used intensively in the PDE theory. The main theorem in this paper states that if $p(x_1,...,x_n)$ is a homogeneous H-Stable polynomial of degree $n$ with nonnegative coefficients; $deg_{p}(i)$ is the maximum degree of the variable $x_i$, $C_i = \min(deg_{p}(i),i)$ and $Cap(p) = \inf_{x_i > 0, 1 \leq i \leq n} {p(x_1,...,x_n)\over x_1 \cdots x_n}$ then the following inequality holds $${\partial^n\over\partial x_1...\partial x_n} p(0,...,0) \geq Cap(p) \prod_{2 \leq i \leq n} \bigg({C_i-1\over C_i}\bigg)^{C_{i}-1}.$$ This inequality is a vast (and unifying) generalization of the Van der Waerden conjecture on the permanents of doubly stochastic matrices as well as the Schrijver-Valiant conjecture on the number of perfect matchings in $k$-regular bipartite graphs. These two famous results correspond to the H-Stable polynomials which are products of linear forms. Our proof is relatively simple and "noncomputational"; it uses just very basic properties of complex numbers and the AM/GM inequality.

Max algebra and the linear assignment problem

Max-algebra, where the classical arithmetic operations of addition and multiplication are replaced by a⊕b:=max(a, b) and a⊗b:=a+b offers an attractive way for modelling discrete event systems and optimization problems in production and transportation. Moreover, it shows a strong similarity to classical linear algebra: for instance, it allows a consideration of linear equation systems and the eigenvalue problem. The max-algebraic permanent of a matrix A corresponds to the maximum value of the classical linear assignment problem with cost matrix A. The analogue of van der Waerden's conjecture in max-algebra is proved. Moreover the role of the linear assignment problem in max-algebra is elaborated, in particular with respect to the uniqueness of solutions of linear equation systems, regularity of matrices and the minimal-dimensional realisation of discrete event systems. Further, the eigenvalue problem in max-algebra is discussed. It is intimately related to the best principal submatrix problem which is finally investigated: Given an integer k, 1≤k≤n, find a (k×k) principal submatrix of the given (n×n) matrix which yields among all principal submatrices of the same size the maximum (minimum) value for an assignment. For k=1,2,...,n, the maximum assignment problem values of the principal (k×k) submatrices are the coefficients of the max-algebraic characteristic polynomial of the matrix for A. This problem can be used to model job rotations.

Bipartite Matching and Van der Waerden Conjecture

In this paper I show that the free energy F and the cost C associated to a bipartite matching problem can be explicitly estimated in term of the solution of a suitable system of equations (cavity equations in the following). The proof of these results relies on a well known result in combinatorics: the Van der Waerden conjecture (Egorychev–Falikman Theorem). Cavity equations, derived by a mean field argument by Mezard and Parisi, can be considered as a smoothed form of the dual formulation for the bipartite matching problem. Moreover cavity equation are the Euler–Lagrange equations of a convex functional G parameterized by the temperature T. In term of their unique solution it is possible to define a free-energy-like function of the temperature g(T). g is a strictly decreasing concave function of T and C=g(0). The convexity of G allows to define an explicit algorithm to find the solution of the cavity equations at a given temperature T. Moreover, once the solution of the cavity equations at a given temperature T is known, the properties of g allow to find exact estimates from below and from above of the cost C.

Solutions of permanental equations regarding stochastic matrices

It is known that the so-called van der Waerden's conjecture, regarding doubly stochastic matrices, was solved in full generality in 1980 and 1981, respectively. In this paper, we deal with equations regarding stochastic matrices generated by double stochastic matrices. Let the quantities t k ( A), ( k = 0, 1, …, n) be defined by (1.1), where A is an n × n doubly stochastic matrix. Moreover, let the system of operators C (Definition 1.1) be given. The results of the paper are the following. If the upper permanent and the lower permanent of two stochastic matrices are equal, then at least one of the factors is equal to A 0, where A 0 is the matrix with entries 1 n (Theorem 1.1). Theorems 1.2–1.4 deal with the means of permanents. An inequality (Lemma 1.2) and a consequence of it (Lemma 1.3) are used in the proofs. These results were obtained when the author was a student in 1932. The paper underlines the importance of the concomitant (Definition 1.2) of a matrix.

Perfect Matchings in $\epsilon$-regular Graphs

A super $(d,\epsilon)$-regular graph on $2n$ vertices is a bipartite graph on the classes of vertices $V_1$ and $V_2$, where $|V_1|=|V_2|=n$, in which the minimum degree and the maximum degree are between $ (d-\epsilon)n$ and $ (d+\epsilon) n$, and for every $U \subset V_1, W \subset V_2$ with $|U| \geq \epsilon n$, $|W| \geq \epsilon n$, $|{{e(U,W) }\over{|U||W|}}-{{e(V_1,V_2)}\over{|V_1||V_2|}}| < \epsilon.$ We prove that for every $1>d >2 \epsilon >0$ and $n>n_0(\epsilon)$, the number of perfect matchings in any such graph is at least $(d-2\epsilon)^n n!$ and at most $(d+2 \epsilon)^n n!$. The proof relies on the validity of two well known conjectures for permanents; the Minc conjecture, proved by Brégman, and the van der Waerden conjecture, proved by Falikman and Egorichev.

On the Asymptotic Solution of a Card-Matching Problem

Introduction Recently S. G. Penrice [8] has shown how an asymptotic result concerning the number of derangements in a classical card-matching problem can be obtained by using two hard celebrated inequalities, known as Minc's conjecture and Van der Waerden's conjecture. Both were verified years after they were proposed. It will be demonstrated here that this result in fact follows from easy inequalities and a straightforward application of the principle of inclusion and exclusion. We show this by proving a more general result. However, as will be explained, the inequalities for the derangement probability established by Penrice yield an asymptotic result that otherwise seems difficult to verify. For a standard deck of cards, with k = 4 suits and n = 13 denominations in each suit, the problem is equivalent to a card game played in Las Vegas: The cards of a shuffled deck are dealt one at a time and face up. At the same time the player calls the denominations in the order ace, two, three,... queen, king; ace, two,... . A match occurs when the player calls the same denomination as the card dealt; the suits need not match. The player wins if no match occurs. If one continues the comparison through the whole deck, the number of matches X is a random variable on the sample space S13x4, which consists of all possible shufflings of the cards. What can we say, to begin, about the derangement probability po = P(X = 0), which is the probability that the player wins? Using the Penrice inequalities (6) below,

Elementary proof for a Van der Waerden's conjecture and related theorems

A well-known conjecture of Van der Waerden says that for the permanent Per A of an n × n doubly stochastic matrix A we have (3.0), with equality if and only if all entries of the matrix A equal to n −1 . In 1977 [1], the author proved that if A is an n × n doubly stochastic matrix, and p ≥ 0, q ≥ 0, p + q = 1, then (2.0) holds with equality if and only if all entries of the matrix A equal to n −1 . In this paper, we show that (3.0) and (2.0) are equivalent. On the basis of this equivalence one can say that the equivalent of the Van der Waerden's conjecture was solved already in 1977. A further subject of the paper is to show similar equivalence theorems concerning permanents of doubly stochastic matrices, moreover a refinement of the Van der Waerden's theorem. A separate section deals with the probabilistic interpretation of some previous results.

Van der Waerden's Conjecture on Permanents

Van der Waerden's Conjecture on Permanents

Theory of permanents 1978–1981

This is a sequel to the author's monograph Permanents a survey of developments in the theory of permanents in the years which followed the publication of the monograph. In particular, it contains Egoryĉev's proof of the van der Waerden conjecture, and a description of the current status of other conjectures. A comprehensive list of papers and books, either published during the period 1978–1981 or awaiting publication, is appended, together with an addenda to the bibliography in the monograph for the years preceding 1978.

The van der waerden conjecture: two proofs in one year

One of the famous open problems in combinatorial theory was the van der Waerden conjecture on permanents of doubly stochastic matrices. After more than fifty years in which it managed to resist attacks it has finally been proved. As so often happens two mathematicians gave independent proofs nearly at the same time. At the end of 1980 the news that G. P. Egoritsjev (F. H. EropbiyeB) [2] had found a proof spread quickly and within a few months translations and expositions of the proof were circulating (cf. D. E. Knuth [5], J. H. van Lint [6]). It came as quite a shock when Matemati~eski Zametki 29 No. 6 appeared a few months ago with a paper by D. I. Falikman ()I. I/I. ~a~lnI<Man) [3], submitted 14.5. 1979 (!), with a completely different proof of the conjecture. Perhaps even more surprising is the fact that the two proofs have as a common feature that they use an elegant inequality, due to A. D. Alexandroff and W. Fenchel, which until quite recently seemed to be unknown to combinatorialists. From the literature it is obvious that geometers certainly know about the inequality. In this note we shall describe and compare the two new proofs and the ideas that led to their discovery. Let A be a square matrix of size n with entries a 6 (1 <~ i, j ~< n). We define the permanent of A (notation: perA) by

On the van der Waerden conjecture and zeros of polynomials

Let F(x) = ∑ k=o n n k A kx k A n ≠ 0, and G(x) = ∑ k=o n n k B kx k B n ≠ 0, be polynomials with real zeros satisfying A n −1 = B n −1 = 0, and let H(x) = ∑ k=o n-2 n k A kB kx k. Using the recently proved validity of the van der Waerden conjecture on permanents, some results on the real zeros of H( x) are obtained. These results are related to classical results on composite polynomials.

Notes on Egoritsjev's proof of the van der Waerden conjecture

A solution to the van der Waerden conjecture on the minimum of the permanent of doubly stochastic matrices has recently been found by G.P. Ergoritsjev (in Russian). Because of the wide interest in this conjecture, it seems useful to make the solution more accessible. Professor van Lint has kindly consented to have his notes (including a direct proof of an inequality available in the literature only in a more general form) published in this journal. Ed . ( R.A.B. )

Proof of the van der Waerden conjecture regarding the permanent of a doubly stochastic matrix

Proof of the van der Waerden conjecture regarding the permanent of a doubly stochastic matrix

On the Đoković conjecture for matrices of rank two

The Đokovic Conjecture asserts that here A is any doubly stochastic n × n matrix and pk (A) is the sum of the permanents of all k × k submatrices of A. We prove that if the case k = l of (∗) holds for all l × l, l  n, doubly stochastic matrices of rank two, then all the other cases of (∗) also hold for the same matrices. In a preceding paper a similar result was obtained for the generalized van der Waerden conjecture. In both cases the proof is based on a representation of p k(A) by means of a function (f(x)g(x)) of a pair of associated polynomials {f(x)g(x)}. Using this representation we also obtain some results on minimizing matrices corresponding to the van der Waerden and to the Dokovic conjectures. Finally we outline some properties of the function (f(x)g(x))

On Some Zero Configurations Associated with the van der Waerden Conjecture

Let A be a permanent minimizing doubly stochastic matrix. This paper discusses the maximum number of zeros which can occur in any row or column of A. The results are applied to reaffirming the van der Waerden conjecture in the cases n⩽4.

On the Van Der Waerden conjecture for matrices of rank two

The generalized van der Waerden conjecture asserts that Here Ais any doubly stochastic n×n matrixJn is the n×n matrix all whose entries are equal to 1/n and pk (A) is the sum of the permanents of all k×k submatrices of A. The casek=n of (*) is the (ordinary) van der Waerden conjecture.It is proved that if the van der Waerden conjecture holds for all k×k k⩽n doubly stochastic matrices of rank two, then the generalized conjecture also holds for the same matrices. The proof is based on a representation of pk (A) by means of an associated pair of polynomials.

A study of the van der Waerden conjecture and its generalizations

It Is shown here that a permanent of n× ndoubly stochastic matrix is not less than 1/ n!. The proof of this inequality follows from the study of a generalized van der Waerden problem on the set of doubly stochastic matrices.