444 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
CAD Adjacency Computation Using Validated Numerics
We present an algorithm for computation of cell adjacencies for well-based
cylindrical algebraic decomposition. Cell adjacency information can be used to
compute topological operations e.g. closure, boundary, connected components,
and topological properties e.g. homology groups. Other applications include
visualization and path planning. Our algorithm determines cell adjacency
information using validated numerical methods similar to those used in CAD
construction, thus computing CAD with adjacency information in time comparable
to that of computing CAD without adjacency information. We report on
implementation of the algorithm and present empirical data.Comment: 20 page
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