179 research outputs found
RealCertify: a Maple package for certifying non-negativity
Let (resp. ) be the field of rational (resp. real)
numbers and be variables. Deciding the non-negativity
of polynomials in over or over semi-algebraic
domains defined by polynomial constraints in is a classical
algorithmic problem for symbolic computation.
The Maple package \textsc{RealCertify} tackles this decision problem by
computing sum of squares certificates of non-negativity for inputs where such
certificates hold over the rational numbers. It can be applied to numerous
problems coming from engineering sciences, program verification and
cyber-physical systems. It is based on hybrid symbolic-numeric algorithms based
on semi-definite programming.Comment: 4 pages, 2 table
Strong bi-homogeneous B\'{e}zout theorem and its use in effective real algebraic geometry
Let f1, ..., fs be a polynomial family in Q[X1,..., Xn] (with s less than n)
of degree bounded by D. Suppose that f1, ..., fs generates a radical ideal, and
defines a smooth algebraic variety V. Consider a projection P. We prove that
the degree of the critical locus of P restricted to V is bounded by
D^s(D-1)^(n-s) times binomial of n and n-s. This result is obtained in two
steps. First the critical points of P restricted to V are characterized as
projections of the solutions of Lagrange's system for which a bi-homogeneous
structure is exhibited. Secondly we prove a bi-homogeneous B\'ezout Theorem,
which bounds the sum of the degrees of the equidimensional components of the
radical of an ideal generated by a bi-homogeneous polynomial family. This
result is improved when f1,..., fs is a regular sequence. Moreover, we use
Lagrange's system to design an algorithm computing at least one point in each
connected component of a smooth real algebraic set. This algorithm generalizes,
to the non equidimensional case, the one of Safey El Din and Schost. The
evaluation of the output size of this algorithm gives new upper bounds on the
first Betti number of a smooth real algebraic set. Finally, we estimate its
arithmetic complexity and prove that in the worst cases it is polynomial in n,
s, D^s(D-1)^(n-s) and the binomial of n and n-s, and the complexity of
evaluation of f1,..., fs
Computing the dimension of real algebraic sets
Let be the set of real common solutions to in
and be the maximum total degree of the
's. We design an algorithm which on input computes the dimension of
. Letting be the evaluation complexity of and , it runs using
arithmetic operations in and
at most isolations of real roots of polynomials of degree at most
. Our algorithm depends on the real geometry of ; its practical
behavior is more governed by the number of topology changes in the fibers of
some well-chosen maps. Hence, the above worst-case bounds are rarely reached in
practice, the factor being in general much lower on practical
examples. We report on an implementation showing its ability to solve problems
which were out of reach of the state-of-the-art implementations.Comment: v2: title chang
Computing roadmaps in smooth real algebraic sets
International audienceLet (f1, . . . , fs) be polynomials in Q[X 1 , . . . , Xn ] of degree bounded by D that generate a radical equidimensional ideal of dimension d and let V ⊂ C^n be the locus of their complex zero set which is supposed to be smooth. A roadmap in V ∩ R^n is a real algebraic curve contained in V ∩ Rn which has a non-empty and connected intersection with each connected component of V ∩ R^n . The classical strategy to compute roadmaps is due to J. Canny and leads to algorithms having a complexity within D^O(n^2) arithmetic operations in Q. This strategy is based on computing a polar variety of dimension 1 and a recursion on the studied variety intersected with fibers taken above a critical value of a projection. Thus, it requires computations with real algebraic numbers and introduces singularities at each recursive call. Thus, no efficient implementation of roadmap algorithms have been obtained until now. Our aim is to provide an efficient implementation of the roadmap algorithm. We show how to slightly modify this strategy in order to avoid the use of real algebraic numbers and to deal with smooth algebraic sets at each recursive call in the case where the input variety is smooth. Our complexity is h^d D^O(n) operations in Q where h bounds the number of recursive call in our algorithm. This quantity is related to the geometry of V ∩ R^n and is bounded by D^O(n), thus in worst cases our algorithm has a complexity within D^O(n^2) arithmetic operations. We report on some experiments done with a preliminary implementation of our algorithm
Testing Sign Conditions on a Multivariate Polynomial and Applications
Let be a polynomial in \Q[X_1, \ldots, X_n] of degree . We focus on testing the emptiness and computing at least one point in each connected component of the semi-algebraic set defined by (or or ). To this end, the problem is reduced to computing at least one point in each connected component of a hypersurface defined by for e\in \Q {\em positive and small enough}. We provide an algorithm allowing us to determine a positive rational number which is small enough in this sense. This is based on the efficient computation of the set of {\em generalized critical values} of the mapping f: y\in \C^n \rightarrow f(y)\in \C which is the union of the classical set of critical values of the mapping and of {\em asymptotic critical values} of the mapping . Then, we show how to use the computation of generalized critical values in order to obtain an efficient algorithm deciding the emptiness of a semi-algebraic set defined by a single inequality or a single inequation. At last, we show how to apply our contribution to determining if a hypersurface contains real regular points. We provide complexity estimates for probabilistic versions of the latter algorithms which are within arithmetic operations in \Q. The paper ends with practical experiments showing the efficiency of our approach
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