Mohab Safey El Din
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View article: Refined bit complexity for the computation of at least onepoint per connected component of a smooth completeintersection real algebraic set
Refined bit complexity for the computation of at least onepoint per connected component of a smooth completeintersection real algebraic set Open
We refine the bit complexity analysis of an algorithm for the computation of at least one point per connected component of a smooth real algebraic set, yielding exponential speedup (with respect to the number of variables) compared to prio…
View article: Probabilistic algorithm for computing all local minimizers of Morse functions on a compact domain
Probabilistic algorithm for computing all local minimizers of Morse functions on a compact domain Open
Let K be the unit-cube in Rn and f\,: K $\rightarrow$ R^n be a Morse function. We assume that the function f is given by an evaluation program $Γ$ in the noisy model, i.e., the evaluation program $Γ$ takes an extra parameter $η$ as input a…
View article: Solving generic parametric linear matrix inequalities
Solving generic parametric linear matrix inequalities Open
We consider linear matrix inequalities (LMIs) $A = A_0 + x_1 A_1 + ... + x_n A_n \succeq 0$ with the $A_i$'s being $m \times m$ symmetric matrices, with entries in a ring $\mathcal{R}$. When $\mathcal{R} = \mathbb{R}$, the feasibility prob…
View article: Optimized Gröbner basis algorithms for maximal determinantal ideals and critical point computations
Optimized Gröbner basis algorithms for maximal determinantal ideals and critical point computations Open
Given polynomials and 1 , . . ., , all in k[ 1 , . . ., ] for some field k, we consider the problem of computing the critical points of the restriction of to the variety defined by 1 = • • • = = 0.These are defined by the simultaneous vani…
View article: Solving parameter-dependent semi-algebraic systems
Solving parameter-dependent semi-algebraic systems Open
10 pages
View article: Algebraic Proofs of Path Disconnectedness using Time-Dependent Barrier Functions
Algebraic Proofs of Path Disconnectedness using Time-Dependent Barrier Functions Open
Two subsets of a given set are path-disconnected if they lie in different connected components of the larger set. Verification of path-disconnectedness is essential in proving the infeasibility of motion planning and trajectory optimizatio…
View article: Optimized Gröbner basis algorithms for maximal determinantal ideals and critical point computations
Optimized Gröbner basis algorithms for maximal determinantal ideals and critical point computations Open
Given polynomials $g$ and $f_1,\dots,f_p$, all in $\Bbbk[x_1,\dots,x_n]$ for some field $\Bbbk$, we consider the problem of computing the critical points of the restriction of $g$ to the variety defined by $f_1=\cdots=f_p=0$. These are def…
View article: Solving parameter-dependent semi-algebraic systems
Solving parameter-dependent semi-algebraic systems Open
We consider systems of polynomial equations and inequalities in $\mathbb{Q}[\boldsymbol{y}][\boldsymbol{x}]$ where $\boldsymbol{x} = (x_1, \ldots, x_n)$ and $\boldsymbol{y} = (y_1, \ldots,y_t)$. The $\boldsymbol{y}$ indeterminates are cons…
View article: Determination of All Stable and Unstable Equilibria for Image-Point-Based Visual Servoing
Determination of All Stable and Unstable Equilibria for Image-Point-Based Visual Servoing Open
International audience
View article: Stability analysis of a bacterial growth model through computer algebra
Stability analysis of a bacterial growth model through computer algebra Open
We describe microbial growth and production of value-added chemical compounds in a continuous bioreactor through a dynamical system and we study the local stability of the equilibrium of interest by means of the classical Routh–Hurwitz cri…
View article: Faster real root decision algorithm for symmetric polynomials
Faster real root decision algorithm for symmetric polynomials Open
In this paper, we consider the problem of deciding the existence of real solutions to a system of polynomial equations having real coefficients, and which are invariant under the action of the symmetric group. We construct and analyze a Mo…
View article: A Direttissimo Algorithm for Equidimensional Decomposition
A Direttissimo Algorithm for Equidimensional Decomposition Open
We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. At the core of this algorithm, we combine ideas from the t…
View article: Algebraic certificates for the truncated moment problem
Algebraic certificates for the truncated moment problem Open
The truncated moment problem consists of determining whether a given finitedimensional vector of real numbers y is obtained by integrating a basis of the vector space of polynomials of bounded degree with respect to a non-negative measure …
View article: Fast Algorithms for Discrete Differential Equations
Fast Algorithms for Discrete Differential Equations Open
Discrete Differential Equations (DDEs) are functional equations that relate polynomially a power series $F(t,u)$ in $t$ with polynomial coefficients in a "catalytic" variable $u$ and the specializations, say at $u=1$, of $F(t,u)$ and of so…
View article: Refined $F_5$ Algorithms for Ideals of Minors of Square Matrices
Refined $F_5$ Algorithms for Ideals of Minors of Square Matrices Open
We consider the problem of computing a grevlex Gröbner basis for the set $F_r(M)$ of minors of size $r$ of an $n\times n$ matrix $M$ of generic linear forms over a field of characteristic zero or large enough. Such sets are not regular seq…
View article: Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational Coefficients
Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational Coefficients Open
Assessing non-negativity of multivariate polynomials over the reals, through\nthe computation of {\\em certificates of non-negativity}, is a topical issue in\npolynomial optimization. This is usually tackled through the computation of\n{\\…
View article: Deciding Cuspidality of Manipulators through Computer Algebra and Algorithms in Real Algebraic Geometry
Deciding Cuspidality of Manipulators through Computer Algebra and Algorithms in Real Algebraic Geometry Open
Cuspidal robots are robots with at least two inverse kinematic solutions that\ncan be connected by a singularity-free path. Deciding the cuspidality of\ngeneric 3R robots has been studied in the past, but extending the study to\nsix-degree…
View article: Faster Change of Order Algorithm for Gröbner Bases under Shape and Stability Assumptions
Faster Change of Order Algorithm for Gröbner Bases under Shape and Stability Assumptions Open
Solving zero-dimensional polynomial systems using Gröbner bases is usually done by, first, computing a Gröbner basis for the degree reverse lexicographic order, and next computing the lexicographic Gröbner basis with a change of order algo…
View article: Exact SOHS Decompositions of Trigonometric Univariate Polynomials with Gaussian Coefficients
Exact SOHS Decompositions of Trigonometric Univariate Polynomials with Gaussian Coefficients Open
Certifying the positivity of trigonometric polynomials is of first importance\nfor design problems in discrete-time signal processing. It is well known from\nthe Riesz-Fej\\'ez spectral factorization theorem that any trigonometric\nunivari…
View article: On the degree of varieties of sum of squares
On the degree of varieties of sum of squares Open
We study the problem of how many different sums of squares decompositions a general polynomial $f$ with SOS-rank $k$ admits. We show that there is a link between the variety $\mathrm{SOS}_k(f)$ of all SOS-decompositions of $f$ and the orth…
View article: Globally Optimal Solution to Inverse Kinematics of 7DOF Serial Manipulator
Globally Optimal Solution to Inverse Kinematics of 7DOF Serial Manipulator Open
The Inverse Kinematics (IK) problem is to nd robot control parameters to\nbring it into the desired position under the kinematics and collision\nconstraints. We present a global solution to the optimal IK problem for a\ngeneral serial 7DOF…