Real Topological Zeta Functions

• Real topological zeta functions are rational invariants in Q(s) that encode the geometry of real singularities through resolution data and blow-Nash equivalence.
• They are constructed from embedded resolutions using combinatorial and convex-geometric techniques to capture subtle invariants like real intersection behavior and pole classification.
• Applications include classifying singularities, linking poles to Bernstein–Sato polynomial roots, and providing robust computational frameworks for analyzing real algebraic varieties.

Real topological zeta functions are rational functions $\in \mathbb{Q}(s)$ associated to a real polynomial $f \in \mathbb{R}[x_1, \dots, x_d]$, providing deep combinatorial invariants of singularity geometry, resolution data, and blow-Nash equivalence. These functions generalize the Denef–Loeser topological zeta functions from the complex to the real algebraic category and form specializations of real motivic zeta functions in the Grothendieck ring of real algebraic varieties. Real topological zeta functions encode subtle invariants of real singularities and are central to modern approaches in singularity theory, motivic integration, and the study of blow-Nash equivalence.

1. Definition and Construction

Let $f \in \mathbb{R}[x_1, \dots, x_d]$ with $f(0) = 0$. Construct an embedded resolution of singularities $$\sigma \colon (X, \sigma^{-1}(0)) \to (\mathbb{R}^d, 0)$$, typically via a finite sequence of real blowings-up or a blow-Nash modification. The total transform decomposes as $$\operatorname{Div}(f \circ \sigma) = \sum_{j \in J} N_j E_j$$ and $K_X = \sum_{j \in J} (\nu_j - 1) E_j$, where the $E_j$ are irreducible components, $N_j$ is the multiplicity of $f \circ \sigma$ along $E_j$, and $\nu_j - 1$ is the multiplicity of $\operatorname{Jac}\sigma$ along $E_j$.

The real topological zeta function is defined by

$$Z_{top,0}(f; s) = \sum_{\varnothing \neq I \subset J} \mu(E_I^\circ \cap \sigma^{-1}(0)(\mathbb{R})) \prod_{i \in I} \frac{1}{\nu_i + N_i s} \in \mathbb{Q}(s)$$

where $E_I^\circ$ is the canonical stratum for $I \subset J$, and $\mu(X) = \beta(X)|_{u=1}$ with $\beta$ the virtual Poincaré polynomial, acting as an “additive invariant” on real algebraic sets.

Associated signed topological zeta functions $Z_{top,0}^{\pm}(f; s)$ arise analogously, replacing the stratum with its corresponding $m$-th root covering $\widetilde{E}_I^{0,\pm}$. These rational functions depend only on the geometry of the resolution and the underlying real singularity structure, and, crucially, are invariants under blow-Nash equivalence (Jaudon, 5 Jan 2026).

2. Motivic Origins and Specialization Procedures

Real topological zeta functions are obtained by specialization from real motivic zeta functions: $$Z_{mot,0}(f; T) = \sum_{n \geq 1} [\{ \gamma \in \mathcal{L}_n(\mathbb{R}^d, 0) \mid \operatorname{ord}_t(f \circ \gamma) = n \}] \mathbb{L}^{-nd} T^n \in \mathcal{M}_{\mathbb{R}}[[T]]$$ using the Grothendieck ring $K_0(\operatorname{Var}_\mathbb{R})$ and virtual Poincaré polynomial $$\beta: K_0(\operatorname{Var}_\mathbb{R}) \rightarrow \mathbb{Z}[u]$$, followed by the specialization $u \to 1$.

This process ensures that $Z_{top,0}(f; s)$ encodes the topological complexity of $f$’s vanishing locus over $\mathbb{R}$, and the rational expressions for the motivic and topological zeta functions mirror the Denef–Loeser formulas originally developed for the complex case (Jaudon, 5 Jan 2026, Rossmann, 2014).

3. Classification of Poles and Residue Criteria

For $d = 2$ and $f$ a plane real polynomial, the poles of $Z_{top,0}(f; s)$ are determined by the resolution data. Every pole is of the form $s_0 = -\frac{\nu_i}{N_i}$ for some $i$ such that $E_i$ has a nontrivial real locus. The order of a pole is at most two, with order two precisely when two $E_i$, $E_j$ satisfy $\nu_i/N_i = \nu_j/N_j$ and their real loci intersect in a real point.

The residue contribution of each $E_i$ at $s_0$ is given by

$\RTop_i = \frac{1}{N_i} \Big( \mu(E_i^\circ \cap \sigma^{-1}(0)(\mathbb{R})) + \sum_{j \neq i} \mu_{ij} \alpha_j \Big)$

where $\alpha_j = \nu_j - \frac{\nu_i}{N_i} N_j$, and $\mu_{ij}$ quantifies real intersections.

A complete criterion (Theorem A, (Jaudon, 5 Jan 2026)) for pole occurrence is:

• $s_0 = -\frac{1}{N_i}$ for real branches of the strict transform,
• $s_0 = -\frac{\nu_i}{N_i}$ for real exceptional components intersecting at least three real or complex branches.

The signed variants $Z_{top,0}^{\pm}(f; s)$ have poles included in a subset determined by the geometry of $\{f > 0\}$ and may further subject to cancellation scenarios (see Theorem B).

4. Algorithmic Frameworks and Computational Techniques

A convex-geometric approach, originally developed for $p$-adic topological zeta functions and generalized to real-valued contexts, enables algorithmic computation of real topological zeta functions for groups, algebras, and modules. This framework models the zeta function as a sum over regular “toric data,” leveraging Newton polyhedra, domain balance, stratum simplification, and controlled reduction via rank conditions on initial forms.

A two-stage algorithm partitions the domain into regular data via combinatorial subdivision (balance/simplify/reduce) and subsequently computes the topological zeta function using mixed volume and Euler characteristic calculations for complex (or real) algebraic strata (Rossmann, 2014). Notably, this method applies verbatim to real local integrals $\int_{\mathbb{R}^n} |f(x)|^s dx$ when $f$ is real-analytic, confirming that topological zeta functions defined via Denef–Loeser specialization are fundamentally suited to real singularities.

5. Interplay with Bernstein–Sato Polynomials and Singularity Invariants

The poles of real topological zeta functions have conjectured and, in certain cases proven, relations to the roots of the Bernstein–Sato polynomial (“$b$-function”) of $f$. For arrangements such as reduced hyperplane configurations in three dimensions, every pole of $Z_{top}(f, s)$ is a root of $b_f(s)$, a fact established using combinatorial stratification, explicit formulae, and reduction to quotient singularities (Budur et al., 2010). In this context, the real locus and resolution-theoretic invariants align closely with theory and calculation in the complex case, but with modulations due to real intersection loci and associated multiplicities.

Topological zeta functions also encode monodromy information, and, by established conjectures, poles correspond to eigenvalues of local Milnor monodromy, connecting motivic and topological invariants with analytic and discrete invariants of the singularities.

6. Applications, Examples, and Scope

Examples of real topological zeta functions include the calculation for the cusp $f(x, y) = y^2 - x^3$, which yields

$$Z_{top,0}(f; s) = \frac{5 + 4s}{(1 + s)(5 + 6s)}, \quad Z_{top,0}^+(f; s) = \frac{6s + 7}{(1 + s)(5 + 6s)}, \quad Z_{top,0}^-(f; s) = \frac{2s + 3}{(1 + s)(5 + 6s)}$$

with poles at $s = -1$ and $s = -5/6$. For the monomial $x^3 y^4$, $Z_{top,0}(x^3 y^4; s) = \frac{1}{(1 + 3s)(1 + 4s)}$, so the poles are $-1/3$ and $-1/4$.

The computational methods extend to nilpotent groups and higher-rank algebraic structures, contingent on tractability of the domain decomposition and Euler characteristic evaluation. The main computational bottleneck remains the triangulation and topological evaluation in large dimensions, but the framework is widely applicable within the limits of combinatorial explosion.

7. Blow-Nash Equivalence and Invariance Properties

Real topological zeta functions, as specialized from motivic zeta functions, are invariants of blow-Nash equivalence—a notion fundamental in the classification of real singularities. The rationality and independence from resolution choices ensure that $Z_{top,0}(f, s)$ serves as a robust singularity invariant. Specifically, invariance follows from the functorial passage

$$K_0(\operatorname{Var}_\mathbb{R}) \xrightarrow{\beta} \mathbb{Z}[u] \xrightarrow{u \to 1} \mathbb{Z}$$

and the Denef–Loeser formula, which together guarantee only singularity-theoretic and topological data enter into the computation and final rational expressions of the zeta functions.

The precise connection between the poles and resolution-theoretic quantities in the real category is a subject of active research, with parallels and notable differences from the classical theory in the complex setting. This suggests real topological zeta functions may provide further classification tools for singularities beyond current motivic and analytic frameworks.