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Hyperbolic Polynomials Approach to Van der Waerden/Schrijver-Valiant like Conjectures : Sharper Bounds, Simpler Proofs and Algorithmic Applications [article]

Leonid Gurvits
2006 arXiv   pre-print
This theorem is a vast (and unifying) generalization of as the van der Waerden conjecture on the permanents of doubly stochastic matrices as well Schrijver-Valiant conjecture on the number of perfect matchings  ...  A e-hyperbolic polynomial p is called POS-hyperbolic if roots of vectors X ∈ R^n_+ with nonnegative coordinates are also nonnegative (the orthant R^n_+ belongs to the hyperbolic cone) and p(e) > 0 .  ...  Van der Waerden / Schrijver-Valiant like conjectures and homogeneous polynomials Let Hom(m, n) be the linear space of homogeneous polynomials p(x), x ∈ R m of degree n in m real varibles ; correspondingly  ... 
arXiv:math/0510452v3 fatcat:lhvw74oae5b3pdii3zpbf27yu4

Maximizing Determinants under Matroid Constraints [article]

Vivek Madan, Aleksandar Nikolov, Mohit Singh, Uthaipon Tantipongpipat
2020 arXiv   pre-print
To show the approximation guarantee, we utilize recent works on strongly log-concave polynomials and show new relationships between different convex programs studied for the problem.  ...  Finally, we use the estimation algorithm and sparsity results to give an efficient deterministic approximation algorithm with an approximation guarantee that depends solely on the dimension d.  ...  Hyperbolic polynomials approach to van der waerden/schrijver-valiant like conjectures: Sharper bounds, simpler proofs and algorithmic applications.  ... 
arXiv:2004.07886v1 fatcat:2z4i3t4wujggvpwd4lst7m6uf4