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Finding at least one point in each connected component of a real algebraic set defined by a single equation. (English) Zbl 1009.14010
Let $$P$$ be a $$n$$-variate polynomial with rational coefficients, defining an algebraic subset $$V$$ of the real affine space $$\mathbb{R}^n$$. The paper describes an implementable algorithm which decides whether $$V$$ is empty. Otherwise the algorithm returns at least one (real algebraic) point of each connected component of $$V$$. The authors claim that their algorithm has a low “practical complexity”. However they admit that their algorithm does not meet the best known theoretical complexity bounds, at least in worst case. Nevertheless, there is no doubt that, after a suitable rearrangement, their procedure may compete in efficiency with known theoretical algorithms. In case that $$P$$ is squarefree and $$V$$ is a real hypersurface with only finitely many singularities, their complexity claim seems convincing. In case of infinitely many singularities of $$V$$, the authors use a deformation of the equation $$P$$ and this may spoil considerably the “practical complexity” of their procedure.
The paper extends well known algorithmic critical point methods to the case $$V$$ is not compact. In distinction to already existing, similar research in this field, the authors make no attempt to give definition of “practical complexity” in mathematical terms.

##### MSC:
 14Q10 Computational aspects of algebraic surfaces 65H10 Numerical computation of solutions to systems of equations 14P05 Real algebraic sets
FGb; ISOLATE
Full Text:
##### References:
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