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Left-right and conjugation actions on matrix tuples have received considerable attention in theoretical computer science due to their connections with polynomial identity testing, group isomorphism, and tensor isomorphism. In this paper, we…
In this article, we obtain upper bounds on the number of irreducible factors of some classes of polynomials having integer coefficients, which in particular yield some of the well known irreducibility criteria. For devising our results, we…
Let $p$ be a polynomial in non-commutative variables $x_1,x_2,\ldots,x_n$ with constant term zero over an algebraically closed field $K$. The object of study in this paper is the image of this kind of polynomial over the algebra of upper…
This article presents a strongly polynomial-time algorithm for the general linear programming problem. This algorithm is an implicit reduction procedure that works as follows. Primal and dual problems are combined into a special system of…
We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In…
Probabilistic approach to Boolean matrix factorization can provide solutions robustagainst noise and missing values with linear computational complexity. However,the assumption about latent factors can be problematic in real world…
Motion polynomials (polynomials over the dual quaternions with nonzero real norm) describe rational motions. We present a necessary and sufficient condition for reduced bounded motion polynomials to admit factorizations into monic linear…
We design a deterministic subexponential time algorithm that takes as input a multivariate polynomial $f$ computed by a constant-depth circuit over rational numbers, and outputs a list $L$ of circuits (of unbounded depth and possibly with…
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
If K/k is a function field in one variable of positive characteristic, we describe a general algorithm to factor one-variable polynomials with coefficients in K. The algorithm is flexible enough to find factors subject to additional…
General factors are a generalization of matchings. Given a graph $G$ with a set $\pi(v)$ of feasible degrees, called a degree constraint, for each vertex $v$ of $G$, the general factor problem is to find a (spanning) subgraph $F$ of $G$…
The $2 \rightarrow q$ norm of a matrix $X \in \mathbb{R}^{n \times d}$ is defined as $\lVert X \rVert_{2 \rightarrow q} = \sup_{\lVert v \rVert_2 = 1} \lVert Xv \rVert_q$. We give polynomial-time multiplicative approximation algorithms for…
We identify a recursive structure among factorizations of polynomial values into two integer factors. Polynomials for which this recursive structure characterizes all non-trivial representations of integer factorizations of the polynomial…
We show that one can approximate the least fixed point solution for a multivariate system of monotone probabilistic polynomial equations in time polynomial in both the encoding size of the system of equations and in log(1/\epsilon), where…
Let $c(x)$ be a monic integer polynomial with coefficients $0$ or $1$. Write $c(x) = a(x) b(x)$ where $a(x)$ and $b(x)$ are monic polynomials with non-negative real (not necessarily integer) coefficients. The unfair 0--1 polynomial…
A method of calculating probability values from a system of marginal constraints is presented. Previous systems for finding the probability of a single attribute have either made an independence assumption concerning the evidence or have…
Matrix-valued polynomials in any finite number of freely noncommuting variables that enjoy certain canonical partial convexity properties are characterized, via an algebraic certificate, in terms of Linear Matrix Inequalities and Bilinear…
In this article, we prove some factorization results for several classes of polynomials having integer coefficients, which in particular yield several classes of irreducible polynomials. Such classes of polynomials are devised by imposing…
Let $g(X)$ be a polynomial over a finite field ${\mathbb F}_q$ with degree $o(q^{1/2})$, and let $\chi$ be the quadratic residue character. We give a polynomial time algorithm to recover $g(X)$ (up to perfect square factors) given the…
We provide simple criteria and algorithms for expressing homogeneous polynomials as sums of powers of independent linear forms, or equivalently, for decomposing symmetric tensors into sums of rank-1 symmetric tensors of linearly independent…