Related papers: A note on Solving Parametric Polynomial Systems
We provide a complete description of the ideal that serves as the resultant ideal for n univariate polynomials of degree d. We in particular describe a set of generators of this resultant ideal arising as maximal minors of a set of…
Following the works by Lin et al. (Circuits Syst. Signal Process. 20(6): 601-618, 2001) and Liu et al. (Circuits Syst. Signal Process. 30(3): 553-566, 2011), we investigate how to factorize a class of multivariate polynomial matrices. The…
An algorithm to give an explicit description of all the solutions to any tropical linear system $A\odot x=B\odot x$ is presented. The given system is converted into a finite (rather small) number $p$ of pairs $(S,T)$ of classical linear…
We consider systems of strict multivariate polynomial inequalities over the reals. All polynomial coefficients are parameters ranging over the reals, where for each coefficient we prescribe its sign. We are interested in the existence of…
This work presents a framework for a-posteriori error-estimating algorithms for differential equations which combines the radii polynomial approach with Haar wavelets. By using Haar wavelets, we obtain recursive structures for the matrix…
In this paper, we present a new algorithm and an experimental implementation for factoring elements in the polynomial n'th Weyl algebra, the polynomial n'th shift algebra, and ZZ^n-graded polynomials in the n'th q-Weyl algebra. The most…
We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases.…
A method is presented that reduces the number of terms of systems of linear equations (algebraic, ordinary and partial differential equations). As a byproduct these systems have a tendency to become partially decoupled and are more likely…
We consider the problem of efficiently solving a system of $n$ non-linear equations in ${\mathbb R}^d$. Addressing Smale's 17th problem stated in 1998, we consider a setting whereby the $n$ equations are random homogeneous polynomials of…
Using discrete Morse theory, we give an algorithm that prunes the excess of information in the Taylor resolution and constructs a new cellular free resolution for an arbitrary monomial ideal. The pruned resolution is not simplicial in…
Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in its coefficients.…
We present an algorithm which for any given ideal $I\subseteq\mathbb{K} [x,y]$ finds all elements of $I$ that have the form $f(x) - g(y)$, i.e., all elements in which no monomial is a multiple of $xy$.
Polynomial Systems, or at least their algorithms, have the reputation of being doubly-exponential in the number of variables [Mayr and Mayer, 1982], [Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that number of…
Parametric prediction error methods constitute a classical approach to the identification of linear dynamic systems with excellent large-sample properties. A more recent regularized approach, inspired by machine learning and Bayesian…
We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which…
The resultant of two univariate polynomials is an invariant of great importance in commutative algebra and vastly used in computer algebra systems. Here we present an algorithm to compute it over Artinian principal rings with a modified…
Differential-elimination algorithms apply a finite number of differentiations and eliminations to systems of partial differential equations. For systems that are polynomially nonlinear with rational number coefficients, they guarantee the…
We present a new insight into the systematic generation of minimal solvers in computer vision, which leads to smaller and faster solvers. Many minimal problem formulations are coupled sets of linear and polynomial equations where image…
We address the problem of computing a linear separating form of a system of two bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at the distinct…
The real radical ideal of a system of polynomials with finitely many complex roots is generated by a system of real polynomials having only real roots and free of multiplicities. It is a central object in computational real algebraic…