2021/07/19 by Basu, Saugata, Percival, Sarah
#14F25 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #W6830
paper · doi:10.48550/arxiv.2107.08947
Let R be a real closed field and C the algebraic closure of R. We give an algorithm for computing a semi-algebraic basis for the first homology group, H1(S,\mathbbF), with coefficients in a field \mathbbF, of any given semi-algebraic set S ⊂ Rk defined by a closed formula. The complexity of the algorithm is bounded singly exponentially. It is not known how to compute such a basis for the higher homology groups with singly exponential complexity. As an intermediate step in our algorithm we construct a semi-algebraic subset Γ of the given semi-algebraic set S, such that Hq(S,Γ) = 0 for q=0,1. We relate this construction to a basic theorem in complex algebraic geometry stating that for any affine variety X of dimension n, there exists Zariski closed subsets Z(n-1) ⊃ ⋯ ⊃ Z(1) ⊃ Z(0) with dimC Z(i) ≤ i, and Hq(X,Z(i)) = 0 for 0 ≤ q ≤ i. We conjecture a quantitative version of this result in the semi-algebraic category, with X and Z(i) replaced by closed semi-algebraic sets. We make initial progress on this conjecture by proving the existence of Z(0) and Z(1) with complexity bounded singly exponentially (previously, such an algorithm was known only for constructing Z0).