Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Polynomial-time computing over quadratic maps I: sampling in real
  algebraic sets   Given a quadratic map Q : K^n -> K^k defined over a computable subring D of a
real closed field K, and a polynomial p(Y_1,...,Y_k) of degree d, we consider
the zero set Z=Z(p(Q(X)),K^n) of the polynomial p(Q(X_1,...,X_n)). We present a
procedure that computes, in (dn)^O(k) arithmetic operations in D, a set S of
(real univariate representations of) sampling points in K^n that intersects
nontrivially each connected component of Z. As soon as k=o(n), this is faster
than the standard 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 (due to A.Barvinok) 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)=sum_i Y_i^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 K by infinitesimals and
subsequent limit computation by a novel procedure that utilizes knowledge of an
explicit isomorphism between real algebraic sets.

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The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
10
The categories of this paper are: q-bio
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The categories of this paper are: nucl-th
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The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: acc-phys
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The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
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The categories of this paper are: eess