Semi-Algebraic Functions: Key Concepts

• Semi-Algebraic functions are mappings whose graphs are defined by finite Boolean combinations of polynomial equations and inequalities.
• They exhibit strong stratification properties that enable precise analysis of critical points and subdifferential behavior in optimization.
• Their robust algebraic structure supports advanced approximation schemes and extensions, including regulous functions and semi-algebraic splines.

A semi-algebraic function is a mapping whose graph can be expressed as a finite Boolean combination of sets defined by polynomial equations and inequalities. Semi-algebraic functions generalize polynomial and rational functions, allowing intricate combinations of polynomial constructs while retaining a rich algebraic and geometric structure. These functions underpin much of real algebraic geometry, variational analysis, and optimization theory, thanks to their stability under algebraic operations, stratification properties, and definability in o-minimal structures. Recent research highlights advanced extension of classical results (e.g., Sard’s theorem), refined subdifferential calculus, topological invariants, and approximation techniques, all leveraging the semi-algebraic property.

1. Foundational Concepts of Semi-Algebraic Functions

A subset $S\subset\R^n$ is semi-algebraic if it can be written as a finite union of sets of the form

$\{\,x\in\R^n : P_i(x)=0,\; Q_j(x)<0\,\}$

with $P_i$, $Q_j$ real polynomials. The class is closed under finite unions, intersections, complements, and projections (Tarski–Seidenberg principle) (Drusvyatskiy et al., 2010). A function $f:X \to \R^k$ is semi-algebraic if its graph is a semi-algebraic subset of $\R^n\times \R^k$ (Valette et al., 2015). Semi-algebraic functions include polynomials, rational continuous (regulous) functions with continuous extensions, Euclidean norms, and indicators of regions cut out by polynomial inequalities (Monnier, 2016).

In real closed fields, such as $\R$, Puiseux series fields $k(0^+)$, and certain Hardy fields, the definition and essential properties persist, enabling applications far beyond classical real numerical domains (Valette et al., 2015).

2. Algebraic Structure, Stratification, and Dimension

Every semi-algebraic set admits a finite Whitney stratification into $C^1$ manifolds $M_i$, allowing seamless transitions between algebraic and differential-geometric considerations (Drusvyatskiy et al., 2010). The semi-algebraic dimension of a set $S$ is defined as

$\dim S = \max_i \{ \dim M_i \}$

This provides a robust framework for defining and analyzing critical sets, tangent structures, and measures of size, such as the notion of $(k,v)$-thin sets in real closed fields—where, informally, a set fails to contain balls of positive measure after projection, with the "thinness" governed by convex subgroups of infinitesimals (Valette et al., 2015).

Semi-algebraic functions' stratification underpins essential analytic properties and guarantees that variational or critical phenomena are contained within predictably low-dimensional subsets, crucial for genericity arguments in optimization (Drusvyatskiy et al., 2010).

3. Subdifferential Calculus and Variational Analysis

Let $f:\R^n \to (-\infty, +\infty]$ be a semi-algebraic function. Its graph, subdifferential, and associated set-valued maps (Fréchet, limiting/Mordukhovich, Clarke) are all themselves semi-algebraic and admit dimension and stratification control (Drusvyatskiy et al., 2010): $\dim(\gph \partial f) = n$ This critical result ensures near-generic single-valuedness and stability: for generic parameters, critical points and subdifferential fibers are finite, stable, and move in a locally trivial way (Hardt’s theorem) (Drusvyatskiy et al., 2010).

Quadratic growth conditions—a strong form of local optimality—are strictly equivalent to strong metric subregularity of the subdifferential for semi-algebraic functions (Drusvyatskiy et al., 2013). This equivalence does not generally hold in wider function classes. The stratification and definability properties yield both necessary and sufficient optimality conditions in nonsmooth analysis:

• Necessary: Strong subregularity of the perturbed subdifferential.
• Sufficient: Subgradient bounds and gauge conditions.

4. Extensions: Rational Continuous Functions and Constructibility

The semi-algebraic paradigm has been extended to include rational continuous (regulous) functions, which are continuous extensions of rational functions on real algebraic sets (Monnier, 2016). The ring $SR_0(X)$ of such functions subsumes polynomials and retains the semi-algebraic structure. The Boolean algebra of sign conditions on $SR_0(X)$ coincides with the classical algebra of semi-algebraic subsets. Algebraically constructible functions—those built as finite integer linear combinations of characteristic functions of semi-algebraic sets—can be fully represented as sums of signs of regulous functions: $$A(X) = \left\{ \sum_k \mathrm{sign}(f_k) \mid f_k \in SR^0(X) \right\}$$ Boundary behaviors, polar loci, and Zariski closure of zero sets characterize when regulous opens reduce to polynomial-principal opens, encapsulating deep interactions between topological and algebraic features.

Regulous functions also admit desirable cohomological, bundle-theoretic, and topological (Nullstellensatz, Cartan's Theorems) properties, enriching the function theory for real algebraic varieties (Monnier, 2016).

5. Topology of Semi-Algebraic Functions and Critical Point Theory

A semi-algebraic $C^2$ function $f:X \to \R$ (with $X$ closed semi-algebraic) admits a finite stratification such that the critical points and their indices can be related directly to topological invariants of $X$, including the Euler characteristic and the topology of fibers at infinity (Dutertre, 2010). Index-theoretic formulas generalize classical Morse relations:

• For proper $f$: The sum of critical indices accounts for Euler characteristic changes across fibers.
• For non-proper $f$: Correction terms $A_{f,a}, I_{f,-a}$ capture topology changes from critical points escaping to infinity.

Specialized formulas for linear functions recover and generalize the Gauss-Bonnet theorem and other classical results. The average Poincaré-Hopf index over generic projections recovers the topological measure

$$\chi(X) - \tfrac12 \chi(\mathrm{link}_\infty(X)) = \mathrm{average~index}$$

6. Approximation and Computational Aspects

Semi-algebraic functions admit quantifiably tractable approximation schemes. For a compact domain $\Omega \subset \R^n$, semi-algebraic functions $f:\Omega \to \R$ can be approximated using the Christoffel-Darboux kernel method (Oster et al., 2022). For Lipschitz $f$, the $L^1$ approximation rate is $O(d^{-1/2})$ for polynomial degree $d$ [Marx et al.]. For semi-algebraic or definable $C^k$ regular $f$, the $L^\infty$ rate is

$\|f - q_d\|_{L^\infty(\Omega)} = O(d^{-k}),$

with exact recovery away from a tubular neighborhood around the critical set. Cell decomposition, needle polynomials, and moment kernel bounds form the technical backbone of this approximation theory.

7. Special Classes: Semialgebraic Splines

Semialgebraic splines are piecewise polynomial functions defined over cell complexes whose cells are bounded by real algebraic curves (DiPasquale et al., 2016). They generalize classical polynomial spline theory to domains cut out by polynomial (semi-algebraic) boundaries, forming graded $S=\R[x,y,z]$ modules. The global and local dimension of spline spaces can be computed via Hilbert functions and polynomials, and depend intricately on both degree and smoothness parameters, as well as geometric configuration (e.g., pencil of curves, distinct slopes at vertices). The algebraic structure is fundamentally governed by the ideals generated by powers of edge-defining polynomials.

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Summary Table: Core Properties of Semi-Algebraic Functions

Property	Description	Reference
Definability	Graph as Boolean combination of polynomial equations/inequalities	(Valette et al., 2015)
Stratification	Finite Whitney stratification into $C^1$ manifolds	(Drusvyatskiy et al., 2010)
Subdifferential Dimension	$\dim(\gph \partial f) = n$ for $f:\R^n \to \R$ semi-algebraic	(Drusvyatskiy et al., 2010)
Optimality Conditions	Quadratic growth $\iff$ strong metric subregularity of subdifferential	(Drusvyatskiy et al., 2013)
Topological Invariants	Index-theoretic formula for Euler characteristic with infinity corrections	(Dutertre, 2010)
Approximation Rate	$L^\infty$ rate $O(d^{-k})$ for $C^k$ semi-algebraic $f$	(Oster et al., 2022)
Regulous Function Theory	Extension to continuous extensions of rational functions	(Monnier, 2016)
Spline Module Structure	Graded module/Hilbert polynomial determined by cell complex geometry	(DiPasquale et al., 2016)

Semi-algebraic functions inhabit a mathematically robust setting characterized by algebraic tractability, analytic stratification, and deep connections to topology, optimization, and computational approximation.