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We show that any k-regular bipartite graph with 2n vertices has at least ( (k\Gamma1) k\Gamma1 k k\Gamma2 ) n perfect matchings (1-factors). Equivalently, this is a lower bound on the permanent of any nonnegative integer n \Theta n matrix with each row and column sum equal to k. For any k, the base (k\Gamma1) k\Gamma1 k k\Gamma2 is largest possible. 1.

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We prove that the mixed discriminant of doubly stochastic n-tuples of semidefinite hermitian n × n matrices is bounded below by n! nn and that this bound is uniquely attained at the n-tuple ( 1 1 nI,..., nI). This result settles a conjecture posed by R. Bapat in 1989. We consider various generalizations and applications of this result. 1

We present a randomized approximation algorithm for counting contingency tables, m × n non-negative integer matrices with given row sums R = (r1,..., rm) and column sums C = (c1,..., cn). We define smooth margins (R, C) in terms of the typical table and prove that for such margins the algorithm has quasipolynomial N O(ln N) complexity, where N = r1 + · · · + rm = c1 + · · · + cn. Various classes of margins are smooth, e.g., when m = O(n), n = O(m) and the ratios between the largest and the smallest row sums as well as between the largest and the smallest column sums are strictly smaller than the golden ratio (1 + √ 5)/2 ≈ 1.618. The algorithm builds on Monte Carlo integration and sampling algorithms for logconcave densities, the matrix scaling algorithm, the permanent approximation algorithm, and an integral representation for the number of contingency tables.

Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. Based on this point of view, we introduce a definition of perfect Quantum (operator) matching. We show that the new notion inherits many ”classical ” properties, but not all of them. This new notion goes somewhere beyound matroids. For separable bipartite quantum states this new notion coinsides with the full rank property of the intersection of two corresponding geometric matroids. In the classical situation, permanents are naturally associated with perfects matchings. We introduce an analog of permanents for positive operators, called Quantum Permanent and show how this generalization of the permanent is related to the Quantum Entanglement. Besides many other things, Quantum Permanents provide new rational inequalities necessary for the separability of

Abstract. We give a concise exposition of the elegant proof given recently by Leonid Gurvits for several lower bounds on permanents. 1. PERMANENTS. The permanent of a square matrix A = (ai, j) n i, j=1 is defined by perA = ∑ π∈Sn i=1 n∏ ai,π(i), (1) where Sn denotes the set of all permutations of {1,...,n}. (The name “permanent” has its root in Cauchy’s fonctions symétriques permanentes [2], as a counterpart to fonctions symétriques alternées—the determinants.) Despite its appearance as the simpler twin-brother of the determinant, the permanent has turned out to be much less tractable. Whereas the determinant can be calculated quickly (in polynomial time, with Gaussian elimination), determining the permanent is difficult (“number-P-complete”). As yet, the algebraic behaviour of the permanent function has appeared to a large extent unmanageable, and its algebraic relevance moderate. Most fruitful research on permanents concerns lower and upper bounds for the permanent (see the book of Minc [12]). In this paper we will consider only lower bounds. Indeed, most interest in the permanent function came from the famous van der Waerden conjecture [16] (in fact formulated as a question), stating that the permanent of any n × n doubly stochastic matrix is at least n!/n n, the minimum being attained only by the matrix with all entries equal to 1/n. (Amatrixisdoubly stochastic if it is nonnegative and each row and column sum is equal to 1.) This conjecture was unsolved for over fifty years, which, when contrasted with its simple form, also contributed to the reputation of intractability of permanents. Finally, Falikman [6] and Egorychev [4] were able to prove this conjecture, using a classical inequality of Alexandroff and Fenchel. The proof with eigenvalue techniques also revealed some unexpected nice algebraic behaviour of the permanent function (see, also for background, Knuth [9]andvanLint[10, 11]). Before the proof of the van der Waerden conjecture was found, a weaker conjecture was formulated by Erdős and Rényi [5]. It claims the existence of a real number α3> 1 such that, for each nonnegative integer-valued n × n matrix A with all row and column sums equal to 3, the permanent of A is at least αn 3. This would follow from the van der Waerden conjecture, since 1 A is doubly stochastic, and hence 3 perA = 3 n (

Abstract. We give a concise exposition of the elegant proof given recently by Leonid Gurvits for several lower bounds on permanents. 1. PERMANENTS. The permanent of a square matrix A = (ai, j) n i, j=1 is defined by perA = ∑ π∈Sn i=1 n∏ ai,π(i), (1) where Sn denotes the set of all permutations of {1,...,n}. (The name “permanent” has its root in Cauchy’s fonctions symétriques permanentes [2], as a counterpart to fonctions symétriques alternées—the determinants.) Despite its appearance as the simpler twin-brother of the determinant, the permanent has turned out to be much less tractable. Whereas the determinant can be calculated quickly (in polynomial time, with Gaussian elimination), determining the permanent is difficult (“number-P-complete”). As yet, the algebraic behaviour of the

We derive here the Friedland-Tverberg inequality for positive hyperbolic polynomials. This inequality is applied to give lower bounds for the number of matchings in r-regular bipartite graphs. It is shown that some of these bounds are asymptotically sharp. We improve the known lower bound for the three dimensional monomer-dimer entropy. We present Ryser-like formulas for computations of matchings in bipartite and general graphs. 2000 Mathematics Subject Classification: 05A15, 05A16, 05C70, 05C80, 82B20 Keywords and phrases: Positive hyperbolic polynomials, Friedland-Tverberg inequality, lower bounds for sum of all subpermanents of doubly stochastic matrices of fixed order, lower bounds for matchings, asymptotic lower matching conjecture, monomerdimer partitions and entropies, Ryser-like formulas for matchings. 1