Related papers: Rational Univariate Representations of Bivariate S…
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…
We present an algorithm for computing a separating linear form of a system of bivariate polynomials with integer coefficients, that is a linear combination of the variables that takes different values when evaluated at distinct (complex)…
Isolating the real roots of univariate polynomials is a fundamental problem in symbolic computation and it is arguably one of the most important problems in computational mathematics. The problem has a long history decorated with numerous…
We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method's good…
This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present…
We study the complexity of computing the real solutions of a bivariate polynomial system using the recently proposed algorithm BISOLVE. BISOLVE is a classical elimination method which first projects the solutions of a system onto the $x$-…
We consider dynamical models given by rational ODE systems. Parameter estimation is an important and challenging task of recovering parameter values from observed data. Recently, a method based on differential algebra and rational…
Consider a system of n polynomial equations and r polynomial inequations in n indeterminates of degree bounded by d with coefficients in a polynomial ring of s parameters with rational coefficients of bit-size at most $\sigma$. From the…
This paper is concerned with certifying that a given point is near an exact root of an overdetermined or singular polynomial system with rational coefficients. The difficulty lies in the fact that consistency of overdetermined systems is…
In this contribution, we consider a zero-dimensional polynomial system in $n$ variables defined over a field $\mathbb{K}$. In the context of computing a Rational Univariate Representation (RUR) of its solutions, we address the problem of…
We consider certificates of positivity for univariate polynomials with rational coefficients that are positive over (an interval of)~$\mathbb{R}$. Such certificates take the form of weighted sums of squares (SOS) of polynomials with…
Our probabilistic analysis sheds light to the following questions: Why do random polynomials seem to have few, and well separated real roots, on the average? Why do exact algorithms for real root isolation may perform comparatively well or…
We improve the local generic position method for isolating the real roots of a zero-dimensional bivariate polynomial system with two polynomials and extend the method to general zero-dimensional polynomial systems. The method mainly…
In this paper, a linear univariate representation for the roots of a zero-dimensional polynomial equation system is presented, where the roots of the equation system are represented as linear combinations of roots of several univariate…
We present algorithmic, complexity and implementation results for the problem of isolating the real roots of a univariate polynomial in $B_{\alpha} \in L[y]$, where $L=\QQ(\alpha)$ is a simple algebraic extension of the rational numbers. We…
This paper is about solving polynomial systems. It first recalls how to do that efficiently with a very high probability of correctness by reconstructing a rational univariate representation (rur) using Groebner revlex computation,…
We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem. The first determinantal…
We address univariate root isolation when the polynomial's coefficients are in a multiple field extension. We consider a polynomial $F \in L[Y]$, where $L$ is a multiple algebraic extension of $\mathbb{Q}$. We provide aggregate bounds for…
Based on the partition of parameter space, two algorithms for computing the rational univariate representation of zero-dimensional ideals with parameters are presented in the paper. Unlike the rational univariate representation of…
We present an algorithm for isolating the roots of an arbitrary complex polynomial $p$ that also works for polynomials with multiple roots provided that the number $k$ of distinct roots is given as part of the input. It outputs $k$ pairwise…