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All reports by Author Leonid Gurvits:

TR20-110 | 22nd July 2020
Leonid Gurvits, Jonathan Leake

Capacity Lower Bounds via Productization

Revisions: 1

The purpose of this note is to state and prove a lower bound on the capacity of a real stable polynomial $p(x)$ which is based only on its value and gradient at $x=1$. This result implies a sharp improvement to a similar inequality proved by Linial-Samorodnitsky-Wigderson in 2000. Such inequalities ... more >>>

TR13-141 | 8th October 2013
Leonid Gurvits

Boolean matrices with prescribed row/column sums and stable homogeneous polynomials: combinatorial and algorithmic applications

Revisions: 1

We prove a new efficiently computable lower bound on the coefficients of stable homogeneous polynomials and present its algorthmic and combinatorial applications. Our main application is the first poly-time deterministic algorithm which approximates the partition functions associated with
boolean matrices with prescribed row and (uniformly bounded) column sums within simply ... more >>>

TR11-169 | 14th December 2011
Leonid Gurvits

Unleashing the power of Schrijver's permanental inequality with the help of the Bethe Approximation

Revisions: 2

Let $A \in \Omega_n$ be doubly-stochastic $n \times n$ matrix. Alexander Schrijver proved in 1998 the following remarkable inequality
\begin{equation} \label{le}
per(\widetilde{A}) \geq \prod_{1 \leq i,j \leq n} (1- A(i,j)); \widetilde{A}(i,j) =: A(i,j)(1-A(i,j)), 1 \leq i,j \leq n
We prove in this paper the following generalization (or just clever ... more >>>

TR07-037 | 2nd February 2007
Leonid Gurvits

Polynomial time algorithms to approximate mixed volumes within a simply exponential factor

Revisions: 1

We study in this paper randomized algorithms to approximate the mixed volume of well-presented convex compact sets.
Our main result is a poly-time algorithm which approximates $V(K_1,...,K_n)$ with multiplicative error $e^n$ and
with better rates if the affine dimensions of most of the sets $K_i$ are small.\\
Our approach is ... more >>>

TR06-025 | 19th January 2006
Leonid Gurvits

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

Let $p(x_1,...,x_n) = p(X) , X \in R^{n}$ be a homogeneous polynomial of degree $n$ in $n$ real variables ,
$e = (1,1,..,1) \in R^n$ be a vector of all ones . Such polynomial $p$ is
called $e$-hyperbolic if for all real vectors $X \in R^{n}$ the univariate polynomial
equation ... more >>>

TR05-103 | 17th August 2005
Leonid Gurvits

A proof of hyperbolic van der Waerden conjecture : the right generalization is the ultimate simplification

Consider a homogeneous polynomial $p(z_1,...,z_n)$ of degree $n$ in $n$ complex variables .
Assume that this polynomial satisfies the property : \\

$|p(z_1,...,z_n)| \geq \prod_{1 \leq i \leq n} Re(z_i)$ on the domain $\{(z_1,...,z_n) : Re(z_i) \geq 0 , 1 \leq i \leq n \}$ . \\

We prove that ... more >>>

TR04-070 | 22nd June 2004
Leonid Gurvits

Combinatorial and algorithmic aspects of hyperbolic polynomials

Let $p(x_1,...,x_n) =\sum_{ (r_1,...,r_n) \in I_{n,n} } a_{(r_1,...,r_n) } \prod_{1 \leq i \leq n} x_{i}^{r_{i}}$
be homogeneous polynomial of degree $n$ in $n$ real variables with integer nonnegative coefficients.
The support of such polynomial $p(x_1,...,x_n)$
is defined as $supp(p) = \{(r_1,...,r_n) \in I_{n,n} : a_{(r_1,...,r_n)} \neq 0 ... more >>>

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