Developing an algorithm for computing the Betti numbers of semi-algebraic sets with singly exponential complexity has been a holy grail in algorithmic semi-algebraic geometry and only partial results are known. In this paper we consider the more general problem of computing the image under the homology functor of a continuous semi-algebraic map \(f:X \rightarrow Y\) between closed and bounded semi-algebraic sets. For every fixed \(\ell \ge 0\) we give an algorithm with singly exponential complexity that computes bases of the homology groups \(\text{ H}_i(X), \text{ H}_i(Y)\) (with rational coefficients) and a matrix with respect to these bases of the induced linear maps \(\text{ H}_i(f):\text{ H}_i(X) \rightarrow \text{ H}_i(Y), 0 \le i \le \ell \). We generalize this algorithm to more general (zigzag) diagrams of continuous semi-algebraic maps between closed and bounded semi-algebraic sets and give a singly exponential algorithm for computing the homology functors on such diagrams. This allows us to give an algorithm with singly exponential complexity for computing barcodes of semi-algebraic zigzag persistent homology in small dimensions.

开发一种计算具有单指数复杂度的半代数集的贝蒂数的算法一直是算法半代数几何中的一个圣杯，但目前只知道部分结果。在本文中，我们考虑在闭有界半代数集之间的连续半代数映射\(f:X \rightarrow Y\) 的同调函子下计算图像的更一般问题。对于每个固定的\(\ell \ge 0\)，我们给出一个具有单指数复杂度的算法，用于计算同源群的基\(\text{ H}_i(X), \text{ H}_i(Y)\)（具有有理系数）和关于归纳线性映射的这些基的矩阵\(\text{ H}_i(f):\text{ H}_i(X) \rightarrow \text{ H}_i(Y) , 0 \le i \le \ell \)。我们将该算法推广到闭半代数集和有界半代数集之间的连续半代数映射的更一般（之字形）图，并给出用于计算此类图上的同调函子的单指数算法。这使我们能够给出一种具有单指数复杂度的算法，用于计算小维度中半代数之字形持久同源性的条形码。