Computing the dimension of real algebraic sets
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Let V be the set of real common solutions to F = (f_1, …, f_s) in ℝ[x_1, …, x_n] and D be the maximum total degree of the f_i's. We design an algorithm which on input F computes the dimension of V. Letting L be the evaluation complexity of F and s=1, it runs using O^∼ (L D^n(d+3)+1 ) arithmetic operations in ℚ and at most D^n(d+1) isolations of real roots of polynomials of degree at most D^n. Our algorithm depends on the real geometry of V; 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 D^nd 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.
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