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Abstract Let$$\mathrm {R}$$be a real closed field. Given a closed and bounded semialgebraic set$$A \subset \mathrm {R}^n$$and semialgebraic continuous functions$$f,g:A \rightarrow \mathrm {R}$$such that$$f^{-1}(0) \subset g^{-1}(0)$$, there exist an integer$$N> 0$$and$$c \in \mathrm {R}$$such that the inequality (Łojasiewicz inequality)$$|g(x)|^N \le c \cdot |f(x)|$$holds for all$$x \in A$$. In this paper, we consider the case whenAis defined by a quantifier-free formula with atoms of the form$$P = 0, P>0, P \in \mathcal {P}$$for some finite subset of polynomials$$\mathcal {P} \subset \mathrm {R}[X_1,\ldots ,X_n]_{\leq d}$$, and the graphs of$$f,g$$are also defined by quantifier-free formulas with atoms of the form$$Q = 0, Q>0, Q \in \mathcal {Q}$$, for some finite set$$\mathcal {Q} \subset \mathrm {R}[X_1,\ldots ,X_n,Y]_{\leq d}$$. We prove that the Łojasiewicz exponent in this case is bounded by$$(8 d)^{2(n+7)}$$. Our bound depends ondandnbut is independent of the combinatorial parameters, namely the cardinalities of$$\mathcal {P}$$and$$\mathcal {Q}$$. The previous best-known upper bound in this generality appeared inP. Solernó, Effective Łojasiewicz Inequalities in Semi-Algebraic Geometry, Applicable Algebra in Engineering, Communication and Computing (1991)and depended on the sum of degrees of the polynomials defining$$A,f,g$$and thus implicitly on the cardinalities of$$\mathcal {P}$$and$$\mathcal {Q}$$. As a consequence, we improve the current best error bounds for polynomial systems under some conditions. Finally, we prove a version of Łojasiewicz inequality in polynomially bounded o-minimal structures. We prove the existence of a common upper bound on the Łojasiewicz exponent for certain combinatorially defined infinite (but not necessarily definable) families of pairs of functions. This improves a prior result of Chris Miller (C. Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic (1994)).
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This content will become publicly available on December 3, 2025
On the asymptotic normality of persistent Betti numbers
Abstract Persistent Betti numbers are a major tool in persistent homology, a subfield of topological data analysis. Many tools in persistent homology rely on the properties of persistent Betti numbers considered as a two-dimensional stochastic process$$ (r,s) \mapsto n^{-1/2} (\beta^{r,s}_q ( \mathcal{K}(n^{1/d} \mathcal{X}_n))-\mathbb{E}[\beta^{r,s}_q ( \mathcal{K}( n^{1/d} \mathcal{X}_n))])$$. So far, pointwise limit theorems have been established in various settings. In particular, the pointwise asymptotic normality of (persistent) Betti numbers has been established for stationary Poisson processes and binomial processes with constant intensity function in the so-called critical (or thermodynamic) regime; see Yogeshwaranet al.(Prob. Theory Relat. Fields167, 2017) and Hiraokaet al.(Ann. Appl. Prob.28, 2018). In this contribution, we derive a strong stabilization property (in the spirit of Penrose and Yukich,Ann. Appl. Prob.11, 2001) of persistent Betti numbers, and we generalize the existing results on their asymptotic normality to the multivariate case and to a broader class of underlying Poisson and binomial processes. Most importantly, we show that multivariate asymptotic normality holds for all pairs (r,s),$$0\le r\le s<\infty$$, and that it is not affected by percolation effects in the underlying random geometric graph.
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- Award ID(s):
- 2015575
- PAR ID:
- 10560621
- Publisher / Repository:
- Applied Probability Trust
- Date Published:
- Journal Name:
- Advances in Applied Probability
- ISSN:
- 0001-8678
- Page Range / eLocation ID:
- 1 to 32
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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