Semi-Algebraic Approximation: Theory & Practice

Updated 5 August 2025

Semi-algebraic approximation is the study of representing and approximating geometric, analytic, and combinatorial objects using finite polynomial equalities and inequalities.
It employs techniques such as cell decomposition, moment-SOS hierarchies, and Christoffel-Darboux kernels to achieve precise local and global approximations.
The approach has significant applications in optimization, control systems, and model theory by ensuring algorithmic efficiency and controlled degree complexity.

Semi-algebraic approximation encompasses the paper, construction, and analysis of procedures for representing, approximating, or analyzing geometric, analytic, or combinatorial objects—such as sets, functions, and maps—by means of semi-algebraic sets and functions, i.e., subsets of real or non-archimedean spaces definable by finitely many polynomial equalities and inequalities. This field operates at the intersection of real algebraic geometry, model theory, optimization, approximation theory, and applications in systems and control, offering both qualitative and quantitative approximation schemes, with precise complexity and structural guarantees.

1. Foundations of Semi-Algebraic Approximation

A set $S \subset \mathbb{R}^n$ is semi-algebraic if it is described by a Boolean combination of polynomial equalities and inequalities. Semi-algebraic functions are real-valued functions whose graphs are semi-algebraic sets. Central questions include:

Approximation of Functions: Can general (e.g., analytic, subanalytic) functions be approximated by semi-algebraic ones with controlled accuracy and complexity?
Approximation of Sets: Can arbitrary sets (analytic, subanalytic, etc.) be locally or globally approximated by semi-algebraic and ultimately algebraic sets, preserving dimension and geometric invariants?
Algorithmic and Quantitative Aspects: How efficiently can approximations be constructed, and what guarantees exist on their combinatorial or algebraic complexity?

Modern semi-algebraic approximation draws on tools such as cell decomposition, horn neighborhoods and s-equivalence, moment-sums-of-squares hierarchies, and Christoffel-Darboux kernel methods.

2. Local and Global Approximation of Sets

Local Approximation: For a closed semianalytic set $A \subset \mathbb{R}^n$ at a point $P$ , semi-algebraic approximation typically proceeds via s-equivalence (Ferrarotti et al., 2012, Ferrarotti et al., 2014). Two sets are s-equivalent at $P$ if the Hausdorff distance between their intersections with spheres of radius $r$ centered at $P$ vanishes faster than $r^s$ as $r \to 0$ . The main result confirms that for any $s$ , every such class contains a semi-algebraic (and, for positive codimension, algebraic) representative of the same dimension. This is achieved by Taylor truncation of the defining functions and control via horn neighborhoods.

Dimension-Preserving Approximation: The process can be refined to preserve the local dimension of the set, using a "regular presentation" and recursively constructed polynomial equations: $g_{i+1} = g_{i}^2 - h_{i+1}^m$ with large odd $m$ , ensuring the algebraic set $\{g_{q+1}=0\}$ is s-equivalent to the original set and of the same local dimension (Ferrarotti et al., 2014).

Implications and Limitations:

Every closed semianalytic set (with positive codimension) can be approximated arbitrarily well locally by algebraic sets of the same dimension.
For subanalytic sets that are not semianalytic, or sets with non-isolated singularities, such strong approximation often fails.

3. Approximation of Functions: Efroymson and Model-Theoretic Results

Efroymson’s Theorem and Extensions: Any continuous semi-algebraic function on a smooth semi-algebraic (Nash) manifold admits, for any positive continuous semi-algebraic $\varepsilon$ , a smooth semi-algebraic function $g$ such that $|f-g|<\varepsilon$ pointwise (Valette et al., 2019). This result generalizes to functions definable in any polynomially bounded o-minimal structure admitting $C^0$ cell decomposition, encompassing globally subanalytic functions.

Lipschitz and $C^1$ Approximation: Uniform and controlled approximations are available, preserving Lipschitz constants and first derivatives within prescribed tolerances.

p-adic and Non-Archimedean Contexts: In p-adic semi-algebraic and subanalytic geometry, any locally Lipschitz semi-algebraic function with constant 1 admits a partition into finitely many semi-algebraic cells on which the function is globally Lipschitz with the same constant. Crucially, every such function is approximated on a cell by a fractional monomial: $m(t, y)^b = e(y)\cdot(t-c(y))^a$ together with an additive semi-algebraic center $d(y)$ , with this approximation extending to derivatives (Cluckers et al., 2010).

Model-Theoretic Applications: Approximation via bounded-degree polynomials supports definability results for types in ACVF and structured domination in the model theory of Berkovich spaces. Points of Berkovich analytic spaces over maximally complete fields are in definable bijection with suitable semi-algebraic types, with approximation playing a central role (Poineau, 2012).

4. Quantitative Approximation, Projection, and Choice

Definable Choice and Quantitative Improvements: Classical semi-algebraic triviality permits definable choices (selections in each fiber of a projection), but with extremely high degree complexity. By weakening to approximate choices—selections $\varepsilon$ -close in Hausdorff distance to the original set and to its projection—one obtains representatives with degree linear in the original data and independent of ambient dimension. This improvement leverages the use of algebraic Puiseux series and controlled evaluation maps (Lerario et al., 23 Sep 2024).

Approximation of Projections and Polynomial Images: Given a semi-algebraic set $S$ and a polynomial map $f$ , outer (and sometimes inner) approximations to the image $f(S)$ or the projection $\pi(S)$ are constructed via sequences of polynomial sublevel sets. These are found by means of semidefinite programming relaxations that converge in $L^1$ to the indicator function of the projected set (Magron et al., 2015, Lasserre, 2014). When sets are defined with quantifiers, hierarchies of polynomial approximations provide inner and outer approximations that converge in volume to the original semi-algebraic set.

5. Algorithmic and Moment-SOS Approaches

Volume and Integration via Moment-SOS Hierarchies: The generalized moment problem enables the approximation of Lebesgue measures (volume) or general polynomial integrals over a semi-algebraic set by representing the integral as an optimization over measures, then relaxing this to SDPs (moment-SOS hierarchy) (Tacchi et al., 2019). Sparsity patterns in the description of the set allow for decomposition into low-dimensional subproblems, enabling computation in higher dimensions. Similar ideas are used to approximate the Hausdorff boundary measure (length or surface area) of a basic compact semi-algebraic set by “lifting” the Stokes' formula to a finite-dimensional moment-SOS hierarchy (Lasserre et al., 2020).

Approximating Indicator Functions and Uniform Sample Generation: Outer approximations to semi-algebraic sets by polynomial superlevel sets are constructed by minimizing the $L^1$ -norm of the polynomial over a bounding box, subject to positivity constraints. As the degree increases, these polynomials converge almost everywhere and in $L^1$ to the indicator function. This supports efficient acceptance-rejection uniform sampling algorithms with asymptotically optimal acceptance rates (Dabbene et al., 2014, Dabbene et al., 2015).

Convexification and Inner Approximations: For nonconvex sets, convex inner approximations are built by identifying boundary points where the second fundamental form is non-positive (i.e., local concavity) and adding affine constraints to truncate these regions. This produces tractable convex subsets amenable to convex optimization in control design (Henrion et al., 2011). For star-convex sets, scale-invariant objectives based on the ratio of outer to inner polynomial sublevel set volumes provide a principled way to construct tight approximations (Guthrie, 2022).

6. Function Approximation Beyond Polynomials

Christoffel-Darboux Approximations: Functions—possibly discontinuous or nonsmooth—are approximated with semi-algebraic functions defined as the minimizer $\arg\min_{y} q_{\mu,d}(x,y)$ of a Christoffel-Darboux polynomial built from moments of a measure supported on the graph of the function. This method implicitly handles nonsmoothness and, for semi-algebraic or definable functions, achieves nearly exact (up to a prescribed $\varepsilon$ ) recovery away from critical lower-dimensional sets, thus transitioning from an $L^1$ to an $L^\infty$ rate (Marx et al., 2019, Oster et al., 2022). Explicit rates and uniform approximations are obtained if the function is analytic on finitely many cells.

Best Approximation in Normed Spaces: For closed semi-algebraic sets and with respect to semi-algebraic norms (including strictly convex and differentiable ones), best approximation is unique outside a stratified semi-algebraic hypersurface. For irreducible varieties, the number of critical points of the squared distance function is the degree of a dominant map, which equals the top Chern number in the projective smooth case (Friedland et al., 2013, Friedland et al., 2014). For tensors, such approximation results inform border rank decompositions and uniqueness questions.

7. Proof Complexity and First-Order Theories

Semi-algebraic methods are central to modern proof complexity via sum-of-squares (SoS) and related proof systems. Recent work formalizes constant-degree SoS and polynomial calculus as first-order theories, providing translation and reflection principles and demonstrating that semi-algebraic proof systems capture key combinatorial reasoning, with applications to propositional encodings (e.g., pigeonhole principles) and tight links to polynomial-time approximation algorithms for NP-hard problems (Part et al., 2021).

Summary Table: Key Methods and Notions in Semi-Algebraic Approximation

Method/Concept	Purpose/Domain	Reference
s-Equivalence & Horn Neighborhoods	Local metric approximation of sets	(Ferrarotti et al., 2012, Ferrarotti et al., 2014)
Polynomial Superlevel Sets	Global outer approximation, sampling	(Dabbene et al., 2014, Dabbene et al., 2015)
Moment-SOS Hierarchy	Volume/integral computation, measure approx.	(Tacchi et al., 2019, Lasserre et al., 2020)
Christoffel-Darboux Approach	Uniform approximation of functions (even discontinuous)	(Marx et al., 2019, Oster et al., 2022)
Convexification via Curvature	Convex inner approximation	(Henrion et al., 2011)
Quantitative Approximate Choice	Selection with controlled degree/complexity	(Lerario et al., 23 Sep 2024)

The paper of semi-algebraic approximation encompasses robust theoretical guarantees, practical and algorithmic procedures, and deep links to optimization, model theory, and computational geometry, with ongoing research in improving complexity, extending to broader classes, and unifying theory and computation.