Blow-Nash Equivalence in Real Singularity Theory

• Blow-Nash equivalence is a classification relation on Nash function-germs, defined via real topological and motivic zeta functions that capture singularity data.
• The theory leverages resolution of singularities to derive invariants, using combinatorial formulas that ensure invariance under arc-analytic and blow-Nash mappings.
• Invariants from Blow-Nash equivalence link pole sets of zeta functions to monodromy eigenvalues, refining classical complex singularity invariants within a real setting.

Blow-Nash equivalence is a classification relation on Nash function-germs over real algebraic varieties, grounded in the invariance properties of real motivic zeta functions and their pole sets. The theory is closely tied to topological invariants derived from resolutions of singularities, notably through real topological zeta functions and their specializations, and serves as a real-analog refinement of complex singularity invariants such as those in the Denef-Loeser framework. The blow-Nash equivalence holds at the level of arc-analytic mappings and provides a robust means of distinguishing singularities of real polynomial functions by leveraging their motivic and topological data (Jaudon, 5 Jan 2026).

1. Definition of Real Topological Zeta Functions

Let $f \in \mathbb{R}[x_1, \ldots, x_d]$ with $f(0) = 0$. For an embedded resolution of singularities,

$$\pi: X \rightarrow \mathbb{R}^d, \quad \pi^{-1}(0) = \bigcup_{j \in J} E_j,$$

each $E_j$ is a smooth irreducible hypersurface. The divisors associated to the total transform of $f$ and the Jacobian under $\pi^*$ admit simple normal crossings,

$$\operatorname{div}(f \circ \pi) = \sum_{j \in J} N_j E_j, \quad K_X - \pi^*K_{\mathbb{R}^d} = \sum_{j \in J} (\nu_j - 1) E_j.$$

For nonempty $I \subset J$, define

$$E_I^0 = \left(\bigcap_{i \in I} E_i\right) \setminus \left(\bigcup_{j \notin I} E_j\right).$$

The real local topological zeta function is then

$$Z_{top,0}(f;s) = \sum_{\emptyset \neq I \subset J} \chi(E_I^0 \cap \pi^{-1}(0)) \prod_{i \in I} \frac{1}{\nu_i + s N_i},$$

where $\chi$ is the additive invariant obtained from the virtual Poincaré polynomial at $u = 1$.

Signed variants

$Z^{+}_{top,0}(f;s),\quad Z^{-}_{top,0}(f;s)$

replace $\chi(E_I^0)$ with the Euler-type invariant of a covering $\widetilde E_I^{0,\pm}$, reflecting the sign structure of $f \circ \pi$ on each stratum (Jaudon, 5 Jan 2026).

2. Motivic Origin and Blow-Nash Invariance

These rational functions are specializations of the real motivic zeta functions: $$Z_{mot,0}(f; T) = \sum_{n \geq 1} [\mathcal{X}_n(f)]\,\mathbb{L}^{-nd}\, T^n \in \mathcal{M}_{\mathbb{R}}[[T]],$$ with analogous signed versions. Here, $[\mathcal{X}_n(f)]$ belongs to the Grothendieck ring of real algebraic varieties, $\mathbb{L} = [\mathbb{A}^1]$, and the specialization reflects evaluation at the virtual Poincaré polynomial $\beta$, $u \mapsto 1$.

Motivic zeta functions are invariants of the blow-Nash equivalence of Nash function-germs, and hence so are the associated topological zeta functions. This ensures the invariance of pole sets under blow-Nash equivalence, extending further to arc-analytic classification (Jaudon, 5 Jan 2026).

3. Resolution Formula and Independence

The Denef-Loeser style formula expresses the zeta functions in terms of data from real log-resolutions:

• Naive:

$$Z_{top,0}(f;s) = \sum_{\emptyset \neq I \subset J} \chi(E_I^0 \cap \pi^{-1}(0)) \prod_{i \in I} \frac{1}{\nu_i + s N_i},$$

• Signed:

$$Z^{\pm}_{top,0}(f;s) = \sum_{\emptyset \neq I \subset J} \chi(\widetilde E_I^{0,\pm} \cap \pi^{-1}(0)) \prod_{i \in I} \frac{1}{\nu_i + s N_i},$$

with $\widetilde E_I^{0,\pm} \rightarrow E_I^0$ a locally trivial real covering specified by the sign data.

The combinatorial nature of the formula guarantees its independence from the chosen resolution, grounded in motivic and cohomological arguments. This holds for the naive function, signed versions, and their motivic origins.

4. Pole Classification in Dimension Two

For curves ($f \in \mathbb{R}[x, y]$), the canonical embedded resolution involves dual blow-ups, and intersections of at most two components are relevant. Consequently, any pole has order at most two.

Candidate poles for real divisors $E_j(\mathbb{R}) \neq \emptyset$ are

$-\frac{\nu_i}{N_i}, \qquad i \in J_{\mathbb{R}}.$

Adaptation of Veys’s criterion for the real case yields:

• A double pole $s_0$ occurs only if two distinct real components $E_i, E_j$ intersect in a real point and $\nu_i/N_i = \nu_j/N_j$.
• A simple pole $s_0 = -\nu_i/N_i$ arises if either: (a) $E_i$ is an irreducible strict transform with $E_i(\mathbb{R}) \neq \emptyset$: $s_0 = -1/N_i$; (b) $E_i$ is an exceptional curve with total intersection number at least three: $(E_i \cdot \sum_{j\neq i}E_j) \geq 3$.

Residue contributions vanish for curves with only one or two real intersections, and all nonzero contributions share sign, precluding cancellation. This structure extends analogously to the motivic and signed zeta functions, with potential pole elimination in the signed case if coverings are empty.

5. Blow-Nash Invariance and Complex Comparison

Both $Z_{top,0}(f; s)$ and $Z^{\pm}_{top,0}(f; s)$ are invariants of blow-Nash and arc-analytic equivalence, situating their pole sets as blow-Nash invariants.

Comparisons with Denef-Loeser’s complex topological zeta function

$$Z_{top,0}^\mathbb{C}(f_\mathbb{C}; s) = \sum_{\emptyset \neq I \subset J} \chi(E_I^0(\mathbb{C}) \cap \pi^{-1}(0)) \prod_{i \in I} \frac{1}{\nu_i + s N_i}$$

highlight several phenomena:

• Some complex poles are absent in the real-naive case if the associated stratum lacks real points.
• Pole order may decrease; for example, $x^2 + y^2$ exhibits a double pole $-1$ in the complex case but a simple pole in the real case.
• Signed variants permit further cancellations, as real coverings $\widetilde E_I^{0,+}$ may be trivial or of differing degree, leading to the loss of additional poles.

6. Illustrative Curve-Singularity Examples

Cusp ($f(x, y) = y^2 - x^3$): via three-step resolution,

$Z_{top,0}(f; s) = \frac{5 + 4s}{(1 + s)(5 + 6s)},$

resulting in poles at $s = -1$ and $s = -5/6$. Signed versions persist with the same poles: $$Z^{+}_{top,0}(f; s) = \frac{6s + 7}{(1 + s)(5 + 6s)},\quad Z^{-}_{top,0}(f; s) = \frac{2s + 3}{(1 + s)(5 + 6s)}.$$

Pham-Brieskorn ($f(x, y) = x^{2k} + y^{2k},\; k \geq 2$): blowing-up at the origin yields strict transform without real points,

$Z_{top,0}(f; s) = \frac{1}{1 + k s},$

pole at $s = -1/k$, while the complex case retains an additional pole at $s = -1$.

Polynomial with higher-order factors ($f(x, y) = x\,y\,(x-y)^3(x-2y)^9$): in the signed case, certain real contributions cancel exactly, so $Z^{+}_{top,0}$ possesses strictly fewer poles. Only exceptional components with at least three real intersections, or strict-transform branches with real locus, induce genuine poles.

7. Monodromy Eigenvalue Correspondence

Adapting the Denef-Loeser/A’Campo argument to the real setting, every pole $s_0$ of $Z_{top,0}(f; s)$ (and analogously $Z^{\pm}_{top,0}(f; s)$) corresponds to an eigenvalue $e^{2\pi i s_0}$ for the classical monodromy action on the Milnor fiber at a real point $a$ proximate to the origin on the relevant branch. This generalizes the monodromy conjecture for complex curves to real settings and anchors pole sets to classical topological structures in singularity theory (Jaudon, 5 Jan 2026).