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- W3100802811 abstract "Given a quadratic map $$Q:mathbb{K}^n to mathbb{K}^k $$ defined over a computable subring D of a real closed field $$mathbb{K},$$ and p ∈D[Y1,...,Y k ] of degree d, we consider the zero set $$Z = Z(p(Q(X)), mathbb{K}^n) subseteq mathbb{K}^n$$ of p(Q(X1,...,X n )) ∈D[X1,...,X n ]. We present a procedure that computes, in (dn)O(k) arithmetic operations in D, a set $$mathcal{S}$$ of (real univariate representations of) sampling points in $$mathbb{K}^n$$ that intersects nontrivially each connected component of Z. As soon as k=o(n), this is faster than the known methods that all have exponential dependence on n in the complexity. In particular, our procedure is polynomial-time for constant k. In contrast, the best previously known procedure is only capable of deciding in $$n^{O(k^2 )} $$ operations the nonemptiness (rather than constructing sampling points) of the set Z in the case of p(Y)=∑ i Y 2 and homogeneous Q. A by-product of our procedure is a bound (dn)O(k) on the number of connected components of Z. The procedure consists of exact symbolic computations in D and outputs vectors of algebraic numbers. It involves extending $$mathbb{K}$$ by infinitesimals and subsequent limit computation by a novel procedure that utilizes knowledge of an explicit isomorphism between real algebraic sets." @default.
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- W3100802811 date "2005-04-01" @default.
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- W3100802811 title "Polynomial-time computing over quadratic maps i: sampling in real algebraic sets" @default.
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- W3100802811 doi "https://doi.org/10.1007/s00037-005-0189-7" @default.
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