Effective O-Minimality in Definable Geometry

• Effective o-minimality is a framework that uses computable format and degree invariants to provide explicit bounds for the complexity of definable sets.
• It applies sharply o-minimal structures that ensure polynomial bounds on operations like cell decomposition, connected components, and Betti numbers.
• The framework enables effective quantitative theorems in geometry and number theory by offering recursively bounded estimates and explicit complexity controls.

Effective o-minimality refers to a robust regime within o-minimal geometry in which the combinatorial and topological complexity of definable sets is controlled by explicit, computable invariants—primarily “format” and “degree”—across all logical operations. The concept is realized via sharply o-minimal (“sharp o-minimal” or $\sharp$o-minimal) structures, which support effective bounds for operations such as Boolean combinations, projections, and cell decompositions. This leads to fully explicit complexity estimates in definable geometry and number theory, such as lattice point counting and Betti number bounds, by encoding the logical and algebraic complexity of definable sets into quantifiable parameters. Effective o-minimality stands in marked contrast to the classical o-minimal setting, in which neither the full first-order theory nor the combinatorial invariants admit recursive (algorithmic) axiomatizations or effective bounds in general.

1. Foundational Principles and Definitions

Effective o-minimality is underpinned by the notion of an FD-filtration (Format-Degree filtration) on an o-minimal expansion of the real field. An FD-filtration $\Omega = \{\Omega_{f,D}\}_{f,D \in \mathbb{N}}$ is a doubly-indexed family of subclasses of definable sets such that every definable set belongs to at least one $\Omega_{f,D}$, and the inclusions $$\Omega_{f,D} \subseteq \Omega_{f+1,D} \cap \Omega_{f,D+1}$$ hold. The parameters $f$ (“format”) and $D$ (“degree”) play roles analogous to ambient dimension and total degree in algebraic geometry. For $X \in \Omega_{f,D}$, one defines

$\format(X) = f, \qquad \deg(X) = D.$

Sharp o-minimality is a strengthening of standard o-minimality that attaches explicit control to combinatorial and geometric invariants throughout logical and geometric constructions. A sharply o-minimal pair $(\mathcal{R}, \Omega)$ guarantees not only o-minimality, but also explicit polynomial bounds (in $D$ with coefficients depending only on $f$) on quantities such as the number of connected components of fibers and the number of cells in decompositions (Binyamini et al., 2022, Harrison-Migochi et al., 3 Mar 2025).

Three strengthening regimes are defined:

• Presharp: Pairwise operations (intersection, union, projection, complement) yield format/degree increases controlled by explicit recurrences; connected component counts are bounded polynomially in $D$ for fixed $f$.
• Weakly-sharp: Boolean operations are controlled uniformly in $k$-ary operations.
• Sharp (.5ex-minimal): The most restrictive, all operations result in the smallest possible increase in format and degree.

Reduction and equivalence notions allow abstract comparison and transfer between such filtrations: $\Omega \le \Omega'$ if for all $f$, $D$ there exist explicit transformations on $(f, D)$ taking $\Omega_{f,D} \subseteq \Omega'_{a(f),Q_f(D)}$ for some (explicit) $a(f)$ and $Q_f$.

2. Non-Axiomatizability and Logical Barriers

Classical o-minimality does not lend itself to recursive or effective axiomatizations. For any language $L$ extending the language of real closed fields by at least one new predicate or function, and any recursive (computably enumerable) list of $L$-sentences $\Lambda$, there exists a real closed field $R$ satisfying $\Lambda$ but failing to be pseudo-o-minimal—i.e., not elementarily equivalent to any ultraproduct of o-minimal $L$-structures. This result demonstrates that the first-order theory of o-minimality with parameters (the set of sentences true in all o-minimal $L$-structures) is not recursively enumerable and hence not recursively axiomatizable (Rennet, 2012).

This sharply distinguishes o-minimality from other axiomatizable settings, such as Ax’s theory of pseudo-finite fields or finite linear orders, where effective or recursive axiomatizations do exist. The failure of o-minimality to persist under ultraproducts precludes its theory from being first-order expressible in any effective combinatorial way.

3. Calculus of Format and Degree

Sharp o-minimality delivers a precise calculus for the growth of format and degree under all major logical and geometric operations. If $A \in \Omega_{f,D}$ and $B \in \Omega_{f',D'}$, the following rules represent the format/degree calculus under sharp (.5ex) minimality (Binyamini et al., 2022):

Operation	Output Format	Output Degree
$A^c$ (complement)	$f$	$D$
$\pi_{\ell-1}(A)$ (projection)	$f$	$D$
$A \cup B$, $A \cap B$	$\max(f,f')$	$D + D'$
$A \times \mathbb{R}$	$f+1$	$D$

A first-order formula involving $n$ variables and Boolean combinations of sets $\{X_j\}$ of format $f_j$, degree $D_j$ defines a set in $\Omega_{\max\{n, \max_j f_j\}, \sum_j D_j}$.

The crucial feature is that every definable set, and the output of every allowable operation, can be tracked precisely in terms of $(f, D)$, with constants always effective—i.e., recursively and polynomially bounded.

4. Sharp Cell Decomposition and Topological Complexity

Sharp cell decomposition is the pillar of effective o-minimality. For any collection $X_1, \ldots, X_k$ in $\Omega_{f,D}$ in $I^\ell$, there is a cylindrical cell decomposition into at most $\mathrm{poly}_f(k, D)$ cells, each of format $O_f(1)$ and degree $O_f(D)$. This construction proceeds by controlled induction on the ambient dimension and pairwise sectioning, ensuring no exponential blowup in number or complexity of cells (Binyamini et al., 2022).

The sharp cell decomposition yields immediate topological consequences, notably explicit Betti number bounds: any compact definable $X \subset \mathbb{R}^\ell$ of format $f$, degree $D$ satisfies

$$\sum_{i=0}^\ell b_i(X) \leq \#K \leq \mathrm{poly}_f(D)$$

where $b_i(X)$ are the Betti numbers of $X$ and $K$ is an explicit simplicial complex homeomorphic to $X$, with bounded size and complexity.

5. Effective Quantitative Theorems

The sharply o-minimal framework enables effective, quantitative versions of deep results in o-minimal geometry, such as lattice point counting theorems. For a definable family $Z \subseteq \mathbb{R}^{m+n}$ in a sharply o-minimal structure and full-rank lattice $\Lambda \subset \mathbb{R}^n$, Theorem 3.1 of (Harrison-Migochi et al., 3 Mar 2025) asserts: $$\left| |Z_T \cap \Lambda| - \frac{\operatorname{Vol}(Z_T)}{\det \Lambda} \right| \leq c \sum_{j=0}^{n-1} \frac{V_j(Z_T)}{\lambda_1 \cdots \lambda_j}$$ for every bounded fiber $Z_T$ and a constant $c = \mathrm{poly}_{\mathcal{F}}(D)$, where format $\mathcal{F}$ and degree $D$ are those of $Z$, $V_j$ is the $j$-dimensional volume sum over projections, and $\lambda_1,\ldots,\lambda_n$ are the lattice's successive minima.

All constants in the counting estimate—including cell counts, Davenport constants, and volume-comparison factors—are explicit and polynomial in $D$, with dependence on $\mathcal{F}$ only in coefficients. This effective machinery transfers to settings such as $R_{\exp}$-definable sets via Pfaffian or sub-Pfaffian complexity bounds.

6. Examples and Arithmetic Applications

Instances of sharply o-minimal structures include:

• $$(\mathbb{R}_{\rm alg}, \text{format: ambient dimension, degree: total degree})$$: classic semialgebraic geometry, with effective semialgebraic cell decomposition.
• $(\mathbb{R}_{\rm Pfaff}, \Omega^*)$: restricted Pfaffian structures, achieved via Gabrielov–Vorobjov and Binyamini–Vorobjov results.
• $(\mathbb{R}_{\rm an})$ (the real field with all restricted analytic functions): not even presharp; the failure follows from curves with many intersections, violating uniform component bounds.

The sharpened axioms and decomposition technology have enabled effective uniform bounds in arithmetic geometry—for instance, Wilkie’s conjecture on rational point counts in $\mathbb{R}_{\exp}$-definable sets, leading to explicit, computable bounds on rational points of bounded height (Binyamini et al., 2022).

7. Limitations, Significance, and Further Directions

The non-recursive axiomatizability of o-minimality (Rennet, 2012) precludes a naive “effective theory of o-minimality” as an algorithmically checkable and complete list of first-order axioms. However, by restricting attention to sharply o-minimal frameworks, effective control over complexity is restored for definable sets, enabling explicit bounds in both geometric combinatorics and number theory. A plausible implication is that sharply o-minimal methods form the natural landscape for further effective results in o-minimal geometry, where any theorem provable within classical o-minimality using Boolean combinations, projections, cell decompositions, and volume-comparison arguments can be made effective by systematically working in a sharply o-minimal regime.

The effectiveness hinges on the availability of true “format” and “degree” data associated to definable objects and the preservation of explicit control throughout the logical and geometric operations. This explicit tracking of logical complexity is essential both for theoretical computer science (model theory, complexity theory) and for quantitative questions in arithmetic geometry.

Sharp o-minimality thus provides a complete and effective combinatorial skeleton for many applications traditionally addressed within ordinary o-minimal geometry, offering a resolution to the long-standing gap between definability and effectiveness.