12 research outputs found
Real root finding for equivariant semi-algebraic systems
Let be a real closed field. We consider basic semi-algebraic sets defined
by -variate equations/inequalities of symmetric polynomials and an
equivariant family of polynomials, all of them of degree bounded by .
Such a semi-algebraic set is invariant by the action of the symmetric group. We
show that such a set is either empty or it contains a point with at most
distinct coordinates. Combining this geometric result with efficient algorithms
for real root finding (based on the critical point method), one can decide the
emptiness of basic semi-algebraic sets defined by polynomials of degree
in time . This improves the state-of-the-art which is exponential
in . When the variables are quantified and the
coefficients of the input system depend on parameters , one
also demonstrates that the corresponding one-block quantifier elimination
problem can be solved in time
Computing the Real Isolated Points of an Algebraic Hypersurface
Let be the field of real numbers. We consider the problem of
computing the real isolated points of a real algebraic set in
given as the vanishing set of a polynomial system. This problem plays an
important role for studying rigidity properties of mechanism in material
designs. In this paper, we design an algorithm which solves this problem. It is
based on the computations of critical points as well as roadmaps for answering
connectivity queries in real algebraic sets. This leads to a probabilistic
algorithm of complexity for computing the real isolated
points of real algebraic hypersurfaces of degree . It allows us to solve in
practice instances which are out of reach of the state-of-the-art.Comment: Conference paper ISSAC 202
Real root finding for equivariant semi-algebraic systems
International audienceLet be a real closed field. We consider basic semi-algebraic sets defined by -variate equations/inequalities of symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by . Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by polynomials of degree in time . This improves the state-of-the-art which is exponential in . When the variables are quantified and the coefficients of the input system depend on parameters , one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time
PROBABILISTIC ALGORITHM FOR POLYNOMIAL OPTIMIZATION OVER A REAL ALGEBRAIC SET
Let f,f1,...,fs be polynomials with rational coefficients in the indeterminates X = X1,...,Xn of maximum degree D and V be the set of common complex solutions of F = (f1,...,fs). We give an algorithm which, up to some regularity assumptions on F, computes an exact representation of the global infimum f ⋆ = inf x∈V∩Rnf (x), i.e. a univariate polynomial vanishing at f ⋆ and an isolating interval for f ⋆. Furthermore, this algorithm decides whether f ⋆ is reached and if so, it returns x ⋆ ∈ V ∩Rn such that f (x⋆) = f ⋆. This algorithm is probabilistic. It makes use of the notion of polar varieties. Its complexity is essentially cubic in (sD) n and linear in the complexity of evaluating the input. This fits within the best known deterministic complexity class DO(n). We report on some practical experiments of a first implementation that is available as a Maple package. It appears that it can tackle global optimization problems that were unreachable by previous exact algorithms and can manage instances that are hard to solve with purely numeric techniques. As far as we know, even under the extra genericity assumptions on the input, it is the first probabilistic algorithm that combines practical efficiency with good control of complexity for this problem