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Section: New Results

Algebraic geometry techniques for polynomial systems

Testing the structural stability of $N$-d discrete linear systems

The goal of this work is to propose new computer algebra based methods for testing the structural stability of $N$-d discrete linear systems. Recall that a discrete linear system given by its transfer function $G(z_1,...,z_n)=N(z_1,...,z_n)/D(z_1,...,z_n)$ is said to be stable if and only if the denominator $D(z_1,...,z_n)$ is devoid from zero inside the unit complex poly-disc. This fundamental problem in the analysis of $N$-d systems has been extensively studied these last decades. At the end of the seventies, DeCarlo et al [77] show that testing the previous condition is equivalent to testing the existence of complex zeros on each face of the poly-disc i.e. $D(1,...,z_i,...,1)$ for $i=1...n$ as well as testing the existence of complex zero on the poly-circle i.e. the zeros of $D(z_1,...,z_n)$ when $|z_1|=...=|z_n|=1$.

Starting from the conditions of DeCarlo et al, we propose a new approach that transform the last condition, that is, the non-existence of complex zeros on the unit poly-circle to a condition on the existence of real solutions inside a region of $R^n$. More precisely we propose two type of transformations. The first one reduces the problem to looking for real solutions inside the unit box while the second one reduces the problem to looking for real solutions in the whole space $R^n$. In order to check the existence of real solutions, we use classical computer algebra algorithms for solving systems of polynomial equations. In the case of one or two variables, the appearing systems are generally zero-dimensional. To count or locate the real solutions of such systems, we compute a rational univariate representation [95] , that is a one to one mapping between the solutions of the system and the roots of a univariate polynomial, thus the problem is reduced to a univariate problem. When the number of variables is larger than two, the systems that stem from the conditions above are no longer zero-dimensional. In such case, we use critical points method that allow to compute solutions in each real connected component of the zeros of the systems [65] .

We implemented the previous approach on maple using the external library Raglib  [63] which provides routines for testing the existence of real solutions of an algebraic system. Preliminary tests show the relevance of our approach.

This work is supported by the ANR MSDOS grant.

Efficient algorihtms for solving bivariate algebraic systems

This work addresses the problem of solving a bivariate algebraic system (i.e computing certified numerical approximation of the solutions) via the computation or a rational univariate representation. Such a representation is useful since it allows to turn many queries on the system into queries on univariate polynomials. Given two coprime polynomials $P$ and $Q$ in $Z[x,y]$ of degree bounded by $d$ and bitsize bounded by $\tau$ we present new algorithms for computing rational univariate representation of the system $\{P,Q\}$ and from this representation, isolating the real solutions of $\{P,Q\}$. The cost analysis of these algorithms show that they have a worst-case bit complexity in $sOB(d^6+d^5\tau )$ which improves by a factor $d$ the state-of-the-art complexity.