Hyperbolic polynomials approach to Van der Waerden/Schrijver-Valiant like conjectures: sharper bounds, simpler proofs and algorithmic applications

Let p(x₁,...,x_n) = p(X) , X ∈ Rⁿ be a homogeneous polynomial of degree n in n real variables, e = (1,1,..,1) ∈ Rⁿ be a vector of all ones . Such a polynomial p is called e-hyperbolic if for all real vectors X ∈ Rⁿ the univariate polynomial equation p(te - X) = 0 has all real roots λ₁(X) ≥ ... ≥ λ_n(X). The number of nonzero roots |i :λ_i(X) ≠ 0 | is called Rank_p(X). An e-hyperbolic polynomial p is called POS-hyperbolic if roots of vectors X ∈ Rⁿ₊ with nonnegative coordinates are also nonnegative (the orthant Rⁿ₊ belongs to the hyperbolic cone) and p(e) > 0. Below e₁,...,e_n stands for the canonical orthogonal basis in Rⁿ. The main results of this paper states that if p(x₁,x₂,...,x_n) is a POS-hyperbolic (homogeneous) polynomial of degree n, Rank_p (e_i) = R_i and p(x₁,x₂,...,x_n) ≥ ∏_{1 ≤ i ≤ n} x_i ; x_i > 0, 1 ≤ i ≤ n, then the following inequality holds ∂ⁿ/∂ x₁...∂ x_n p(0,...,0) ≥ ∏_{1 ≤ i ≤ n} (G_i-1/G_i)^Gi_-1, where G_i = min(R_i , n+1-i) . 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 POS-hyperbolic polynomials which are products of linear forms with nonnegative coefficients.Our proof is relatively simple and "noncomputational"; it actually slightly improves Schrijver's lower bound, and uses very basic (more or less centered around Rolle's theorem) properties of hyperbolic polynomials. We present some important algorithmic applications of the result, including a polynomial time deterministic algorithm approximating the permanent of n x n entry-wise non-negative matrices within a multiplicative factor eⁿ/n^m for any fixed positive m; and a deterministic poly-time algorithm approximating the permanent of n x n matrix A having at most k nonzero entries in each column to within a multiplicative factor (k-1/k)^(k-1)n.This paper introduces a new powerful "polynomial" technique , which allows us to simplify and unify hard and key known results as well as to prove new important theorems and get new algorithms.