Related papers: On the Complexity of the Multivariate Resultant
In this paper we give a new and simple algorithm to put any multivariate polynomial into a normal determinant form in which each entry has the form , and in each column the same variable appears. We also apply the algorithm to obtain a…
Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input…
Solving systems of m multivariate quadratic equations in n variables (MQ-problem) over finite fields is NP-hard. The security of many cryptographic systems is based on this problem. Up to now, the best algorithm for solving the underdefined…
We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer…
We consider estimation of the covariance matrix of a multivariate random vector under the constraint that certain covariances are zero. We first present an algorithm, which we call Iterative Conditional Fitting, for computing the maximum…
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that,…
A fundamental problem in computer science is to find all the common zeroes of $m$ quadratic polynomials in $n$ unknowns over $\mathbb{F}_2$. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity…
It is well known that many problems in interval computation are intractable, which restricts our attempts to solve large problems in reasonable time. This does not mean, however, that all problems are computationally hard. Identifying…
The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of…
Given a multivariate real (or complex) polynomial $p$ and a domain $\cal D$, we would like to decide whether an algorithm exists to evaluate $p(x)$ accurately for all $x \in {\cal D}$ using rounded real (or complex) arithmetic. Here…
The Painleve test is very useful to construct not only the Laurent-series solutions but also the elliptic and trigonometric ones. Such single-valued functions are solutions of some polynomial first order differential equations. To find the…
In this paper we obtain complexity bounds for computational problems on algebraic power series over several commuting variables. The power series are specified by systems of polynomial equations: a formalism closely related to weighted…
Polynomial system solving is a classical problem in mathematics with a wide range of applications. This makes its complexity a fundamental problem in computer science. Depending on the context, solving has different meanings. In order to…
The linear complete differential resultant of a finite set of linear ordinary differential polynomials is defined. We study the computation by linear complete differential resultants of the implicit equation of a system of $n$ linear…
Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for…
The Possible-Winner problem asks, given an election where the voters' preferences over the set of candidates is partially specified, whether a distinguished candidate can become a winner. In this work, we consider the computational…
One of the most significant challenges in Computing Determinant of Rectangular Matrices is high time complexity of its algorithm. Among all definitions of determinant of rectangular matrices, used definition has special features which make…
The discriminant of a multivariate polynomial with indeterminate coefficients is not necessarily a hypersurface, and characterizing its codimension was an open problem for quite a while. We resolve this problem for the discriminants of…
In this work, we consider a multivariate regression model with one-sided errors. We assume for the regression function to lie in a general H\"{o}lder class and estimate it via a nonparametric local polynomial approach that consists of…
The aim of this paper is to show an estimate for the determinant of the covariance of a two-dimensional vector of multiple stochastic integrals of the same order in terms of a linear combination of the expectation of the determinant of its…