Complexity and speed of semi-algebraic multi-persistence
Abstract
Let $\mathrm{R}$ be a real closed field, $S \subset \mathrm{R}^n$ a closed and bounded semi-algebraic set and $\mathbf{f} = (f_1,\ldots,f_p):S \rightarrow \mathrm{R}^p$ a continuous semi-algebraic map. We study the poset module structure in homology induced by the simultaneous filtrations of $S$ by the sub-level sets of the functions $f_i$ from an algorithmic and quantitative point of view. For fixed dimensional homology we prove a singly exponential upper bound on the complexity of these modules which are encoded as certain semi-algebraically constructible functions on $\mathrm{R}^p \times \mathrm{R}^p$. We also deduce for semi-algebraic filtrations of bounded complexity, upper bounds on the number of equivalence classes of finite poset modules that such a filtration induces -- establishing a tight analogy with a well-known graph theoretical result on the "speed'' of algebraically defined graphs.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2024
- DOI:
- 10.48550/arXiv.2407.13586
- arXiv:
- arXiv:2407.13586
- Bibcode:
- 2024arXiv240713586B
- Keywords:
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- Mathematics - Algebraic Topology;
- Primary 14F25;
- 55N31;
- Secondary 68W30
- E-Print:
- 35 pages, 1 figure. Comments welcome