A Geometric Index Reduction Method for Implicit Systems of Differential Algebraic Equations

This paper deals with the index reduction problem for the class of quasi-regular DAE systems. It is shown that any of these systems can be transformed to a generically equivalent first order DAE system consisting of a single purely algebraic (polynomial) equation plus an under-determined ODE (that is, a semi-explicit DAE system of differentiation index 1) in as many variables as the order of the input system. This can be done by means of a Kronecker-type algorithm with bounded complexity

Sparse FGLM algorithms

International audienceGiven a zero-dimensional ideal I \subset \kx of degree $D$, the transformation of the ordering of its \grobner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms to perform the change of ordering. The main contributions of this paper are several efficient methods for the change of ordering which take advantage of the sparsity of multiplication matrices in the classical {\sf FGLM} algorithm. Combing all these methods, we propose a deterministic top-level algorithm that automatically detects which method to use depending on the input. As a by-product, we have a fast implementation that is able to handle ideals of degree over $40000$. Such an implementation outperforms the {\sf Magma} and {\sf Singular} ones, as shown by our experiments. First for the shape position case, two methods are designed based on the Wiedemann algorithm: the first is probabilistic and its complexity to complete the change of ordering is $O(D(N_1+n\log (D)))$, where $N_1$ is the number of nonzero entries of a multiplication matrix; the other is deterministic and computes the LEX \grobner basis of $\sqrt{I}$ via Chinese Remainder Theorem. Then for the general case, the designed method is characterized by the Berlekamp--Massey--Sakata algorithm from Coding Theory to handle the multi-dimensional linearly recurring relations. Complexity analyses of all proposed methods are also provided. Furthermore, for generic polynomial systems, we present an explicit formula for the estimation of the sparsity of one main multiplication matrix, and prove its construction is free. With the asymptotic analysis of such sparsity, we are able to show for generic systems the complexity above becomes $O(\sqrt{6/n \pi} D^{2+\frac{n-1}{n}})$

A nearly optimal algorithm for deciding connectivity queries in smooth and bounded real algebraic sets

Major revision, accepted for publication to Journal of the ACMInternational audienceA roadmap for a semi-algebraic set $S$ is a curve which has a non-empty and connected intersection with all connected components of $S$. Hence, this kind of object, introduced by Canny, can be used to answer connectivity queries (with applications, for instance, to motion planning) but has also become of central importance in effective real algebraic geometry, since it is used in higher-level algorithms. In this paper, we provide a probabilistic algorithm which computes roadmaps for smooth and bounded real algebraic sets. Its output size and running time are polynomial in $(nD)^{n\log(d)}$, where $D$ is the maximum of the degrees of the input polynomials, $d$ is the dimension of the set under consideration and $n$ is the number of variables. More precisely, the running time of the algorithm is essentially subquadratic in the output size. Even under our assumptions, it is the first roadmap algorithm with output size and running time polynomial in $(nD)^{n\log(d)}$

Change of order for regular chains in positive dimension

AbstractWe discuss changing the variable order for a regular chain in positive dimension. This quite general question has applications going from implicitization problems to the symbolic resolution of some systems of differential algebraic equations.We propose a modular method, reducing the problem to computations in dimension zero and one. The problems raised by the choice of the specialization points and the lack of the (crucial) information of what are the free and algebraic variables for the new order are discussed. Strong (but not unusual) hypotheses for the initial regular chain are required; the main required subroutines are change of order in dimension zero and a formal Newton iteration