Section: New Results

Solving Polynomial Systems over the Reals and Applications

Probabilistic Algorithm for Computing the Dimension of Real Algebraic Sets

Let $f\in \mathbb{Q}[X_1,...,X_n]$ be a polynomial of degree $D$. We consider the problem of computing the real dimension of the real algebraic set defined by $f=0$. Such a problem can be reduced to quantifier elimination. Hence it can be tackled with Cylindrical Algebraic Decomposition within a complexity that is doubly exponential in the number of variables. More recently, denoting by $d$ the dimension of the real algebraic set under study, deterministic algorithms running in time $D^{O\left(d\right(n-d\left)\right)}$ have been proposed. However, no implementation reflecting this complexity gain has been obtained and the constant in the exponent remains unspecified. In [11] , we design a probabilistic algorithm which runs in time which is essentially cubic in $D^{d\left(n-d\right)}$. Our algorithm takes advantage of genericity properties of polar varieties to avoid computationally difficult steps of quantifier elimination. We also report on a first implementation. It tackles examples that are out of reach of the state-of-the-art and its practical behavior reflects the complexity gain.

Real root finding for determinants of linear matrices

Let $A_0,A_1,...,A_n$ be given square matrices of size m with rational coefficients. The paper [7] focuses on the exact computation of one point in each connected component of the real determinantal variety $\{x\in {ℝ}^n:\det (A_0+x_1{A}_1+\cdots +x_n{A}_n)=0\}$. Such a problem finds applications in many areas such as control theory, computational geometry, optimization, etc. Using standard complexity results this problem can be solved using $m^{O\left(n\right)}$ arithmetic operations. Under some genericity assumptions on the coefficients of the matrices, we provide in an algorithm solving this problem whose runtime is essentially quadratic in ${\left(\genfrac{}{}{0pt}{}{n+m}{n}\right)}^3$. We also report on experiments with a computer implementation of this algorithm. Its practical performance illustrates the complexity estimates. In particular, we emphasize that for subfamilies of this problem where m is fixed, the complexity is polynomial in n.

Real root finding for rank defects in linear Hankel matrices

Let $H_0,...,H_n$ be $m\times m$ matrices with entries in $\mathbb{Q}$ and Hankel structure, i.e. constant skew diagonals. We consider the linear Hankel matrix $H\left(X\right)=H_0+X_1{H}_1+\cdots +X_n{H}_n$ and the problem of computing sample points in each connected component of the real algebraic set defined by the rank constraint $𝗋𝖺𝗇𝗄\left(H\right(X\left)\right)\le r$, for a given integer $r\le m-1$. Computing sample points in real algebraic sets defined by rank defects in linear matrices is a general problem that finds applications in many areas such as control theory, computational geometry, optimization, etc. Moreover, Hankel matrices appear in many areas of engineering sciences. Also, since Hankel matrices are symmetric, any algorithmic development for this problem can be seen as a first step towards a dedicated exact algorithm for solving semi-definite programming problems, i.e. linear matrix inequalities. Under some genericity assumptions on the input (such as smoothness of an incidence variety), we design in [18] a probabilistic algorithm for tackling this problem. It is an adaptation of the so-called critical point method that takes advantage of the special structure of the problem. Its complexity reflects this: it is essentially quadratic in specific degree bounds on an incidence variety. We report on practical experiments and analyze how the algorithm takes advantage of this special structure. A first implementation outperforms existing implementations for computing sample points in general real algebraic sets: it tackles examples that are out of reach of the state-of-the-art.

Optimizing a Parametric Linear Function over a Non-compact Real Algebraic Variety

In [17] , we consider the problem of optimizing a parametric linear function over a non-compact real trace of an algebraic set. Our goal is to compute a representing polynomial which defines a hypersurface containing the graph of the optimal value function. Rostalski and Sturmfels showed that when the algebraic set is irreducible and smooth with a compact real trace, then the least degree representing polynomial is given by the defining polynomial of the irreducible hypersurface dual to the projective closure of the algebraic set. First, we generalize this approach to non-compact situations. We prove that the graph of the opposite of the optimal value function is still contained in the affine cone over a dual variety similar to the one considered in compact case. In consequence, we present an algorithm for solving the considered parametric optimization problem for generic parameters' values. For some special parameters' values, the representing polynomials of the dual variety can be identically zero, which give no information on the optimal value. We design a dedicated algorithm that identifies those regions of the parameters' space and computes for each of these regions a new polynomial defining the optimal value over the considered region.

Bounds for the Condition Number of Polynomials Systems with Integer Coefficients

Polynomial systems of equations are a central object of study in computer algebra. Among the many existing algorithms for solving polynomial systems, perhaps the most successful numerical ones are the homotopy algorithms. The number of operations that these algorithms perform depends on the condition number of the roots of the polynomial system. Roughly speaking the condition number expresses the sensitivity of the roots with respect to small perturbation of the input coefficients. A natural question to ask is how can we bound, in the worst case, the condition number when the input polynomials have integer coefficients? In [19] we address this problem and we provide effective bounds that depend on the number of variables, the degree and the maximum coefficient bitsize of the input polynomials. Such bounds allows to estimate the bit complexity of the algorithms that depend on the separation bound, like the homotopy algorithms, for solving polynomial systems.

Nearly Optimal Refinement of Real Roots of a Univariate Polynomial

In [10] we assume that a real square-free polynomial $A$ has a degree $d$, a maximum coefficient bitsize $\tau$ and a real root lying in an isolating interval and having no nonreal roots nearby (we quantify this assumption). Then, we combine the Double Exponential Sieve algorithm (also called the Bisection of the Exponents), the bisection, and Newton iteration to decrease the width of this inclusion interval by a factor of $t=2^{-L}$. The algorithm has Boolean complexity ${\tilde{O}}_B(d^2\tau +dL)$. Our algorithms support the same complexity bound for the refinement of $r$ roots, for any $r\le d$.

Accelerated Approximation of the Complex Roots and Factors of a Univariate Polynomial

The known algorithms approximate the roots of a complex univariate polynomial in nearly optimal arithmetic and Boolean time. They are, however, quite involved and require a high precision of computing when the degree of the input polynomial is large, which causes numerical stability problems. We observe that these difficulties do not appear at the initial stages of the algorithms, and in [8] we extend one of these stages, analyze it, and avoid the cited problems, still achieving the solution within a nearly optimal complexity estimates, provided that some mild initial isolation of the roots of the input polynomial has been ensured. The resulting algorithms promise to be of some practical value for root-finding and can be extended to the problem of polynomial factorization, which is of interest on its own right. We conclude with outlining such an extension, which enables us to cover the cases of isolated multiple roots and root clusters.

Polynomial Interrupt Timed Automata

Interrupt Timed Automata (ITA) form a subclass of stopwatch automata where reachability and some variants of timed model checking are decidable even in presence of parameters. They are well suited to model and analyze real-time operating systems. Here we extend ITA with polynomial guards and updates, leading to the class of polynomial ITA (polITA). In [13] , we prove that reachability is decidable in 2EXPTIME on polITA, using an adaptation of the cylindrical algebraic decomposition algorithm for the first-order theory of reals using symbolic computation. Compared to previous approaches, our procedure handles parameters and clocks in a unified way. We also obtain decidability for the model checking of a timed version of CTL and for reachability in several extensions of polITA.