Real Motivic Zeta Functions

• Real motivic zeta functions are invariants defined using arc spaces and constructible sets, generalizing topological zeta functions to capture real polynomial singularities.
• They are computed via virtual Poincaré polynomials and embedded resolutions, yielding explicit formulas and candidate poles that reflect numerical data from divisors.
• Both unsigned and signed variants exist, with their pole structures serving as key tools in classifying real singularities under blow–Nash equivalence.

Real motivic zeta functions generalize the topological zeta functions of Denef and Loeser from the complex to the real algebraic setting, associating powerful invariants to real polynomial singularities via arc spaces, constructible sets, and virtual Poincaré polynomials. These constructions provide tools to study blow-Nash equivalence, singularity types, and the distribution of poles, with explicit formulas derivable via embedded resolution of singularities, especially for curves. Both unsigned and signed variants of these functions exist, each encoding subtle real-analytic and topological data on real algebraic sets.

1. Foundations and Definitions

Let $f \in \R[x_1,\ldots,x_d]$ be a real polynomial germ at the origin. Associated to $f$ are the spaces of $n$-jets of real arcs $\mathcal{X}_n(f)$, consisting of arcs $\gamma$ through $0$ with $\operatorname{ord}_t(f\circ\gamma)=n$; $\mathcal{X}_n^{\pm}(f)$ consists of arcs for which the leading coefficient of $f\circ\gamma$ is $\pm1$. These sets are Zariski-constructible, yielding elements of the Grothendieck ring $\mathcal{K}_0(\R\Var)$.

• Naïve real motivic zeta function (unsigned):

$Z_{\mathrm{mot},0}(f;T) = \sum_{n\geq1} [\mathcal{X}_n(f)]\,\LL^{-nd} T^n \in \mathcal{M}_{\mathbb{R}}[[T]]$

where $\LL=[\A^1]$ and $\mathcal{M}_{\R} = \mathcal{K}_0(\R\Var)[\LL^{-1}]$.

• Signed real motivic zeta function:

$Z_{\mathrm{mot},0}^{\pm}(f;T) = \sum_{n\geq1} [\mathcal{X}_n^{\pm}(f)]\,\LL^{-nd} T^n$

• Specialization via the virtual Poincaré polynomial maps classes $X$ via $\beta: \mathcal{K}_0(\R\Var) \to \Z[u]$ by $X \mapsto \sum_i \dim H_i(X;\Z/2) u^i$. For example:

$$Z_{\beta,0}(f;T) = \sum_{n} \beta(\mathcal{X}_n(f)) u^{-nd} T^n$$

Topological real motivic zeta functions are obtained by $T \mapsto u^{-s}$ and $u \to 1$:

$$Z_{\mathrm{top},0}(f;s) = \lim_{u\to1} Z_{\beta,0}(f; u^{-s}),\quad Z_{\mathrm{top},0}^{\pm}(f;s) = \lim_{u\to1} (u-1) Z_{\beta,0}^{\pm}(f;u^{-s})$$

Alternatively, using the additive invariant $\mu := \beta(\cdot)|_{u=1} : \mathcal{K}_0(\R\Var) \to \Z$,

$$Z_{\mathrm{top},0}(f;s) = \sum_{n\ge1} \mu(\mathcal{X}_n(f)) (-1)^{-nd} n^{-s}$$

Both $Z_{\mathrm{top},0}(f;s)$ and $Z_{\mathrm{top},0}^{\pm}(f;s)$ are rational functions in $s$.

2. Invariance and Specialization

The functions $Z_{\mathrm{top},0}$ and $Z_{\mathrm{top},0}^{\pm}$ are specializations of $Z_{\beta,0}$ and $Z_{\beta,0}^\pm$, which themselves are specializations of $Z_{\mathrm{mot},0}$ and $Z_{\mathrm{mot},0}^{\pm}$, respectively. Fichou and Campesato demonstrated these functions are invariants of blow–Nash equivalence (arc-analytic equivalence) of real germs, ensuring that both the functions and their sets of poles are preserved under such equivalence relations (Jaudon, 5 Jan 2026).

This property distinguishes real motivic zeta functions from their complex counterparts and makes them particularly relevant in the classification of real singularities up to blow-Nash equivalence.

3. Explicit Formulation via Resolution of Singularities

For $d=2$, an embedded real algebraic resolution $\sigma: (X, \sigma^{-1}(0)) \to (\R^2,0)$ gives

$$f \circ \sigma = u \prod_{j \in J} y_j^{N_j}, \quad \operatorname{Jac}(\sigma) = v \prod_{j \in J} y_j^{\nu_j-1}$$

with divisors $E_j$, and sets $J_{\R}$ and $J_{\R}^\pm$ corresponding to real divisors with nontrivial real points and those intersecting the closure of $\{f \gtrless 0\}$, respectively.

• Unsigned case:

$$Z_{\mathrm{top},0}(f;s) = \sum_{\emptyset \neq I \subset J_\R} \mu\bigl(E_I^0(\R)\bigr) \prod_{i\in I} \frac{1}{\nu_i + s N_i}$$

In two dimensions, this simplifies to

$$Z_{\mathrm{top},0}(f;s) = \sum_{i \in J_{\R}} \frac{\mu(E_i^0)}{\nu_i + sN_i} + \sum_{i<j,\,i,j\in J_{\R}} \frac{\mu(E_i(\R)\cap E_j(\R))}{(\nu_i+sN_i)(\nu_j+sN_j)}$$

• Signed case:

$$Z_{\mathrm{top},0}^{\pm}(f;s) = \sum_{\emptyset \neq I \subset J^\pm_{\R}} \mu\bigl(\widetilde{E}_I^{0,\pm}(\R)\bigr) \prod_{i\in I} \frac{1}{\nu_i + sN_i}$$

This explicit formula reduces the computation of $Z_{\mathrm{top},0}(f;s)$ for plane curve singularities to topological data associated to the divisors appearing in the resolution.

4. Candidate Poles, Orders, and Criteria

The candidate poles of $Z_{\mathrm{top},0}(f;s)$ and $Z_{\mathrm{top},0}^{\pm}(f;s)$ are the numbers $s = -\nu_i/N_i$ for $i$ in $J_{\R}$ or $J_{\R}^{\pm}$. Each pole has at most order two (Jaudon, 5 Jan 2026).

• Double-pole criterion: A pole $s_0$ is double if and only if two distinct $i \ne j$ with $\nu_i/N_i = \nu_j/N_j = -s_0$ meet in a real point (or in $\overline{\{f\gtrless 0\}}$ for the signed case).
• Simple-pole criterion (unsigned, Jaudon): $s_0 = -\nu_i / N_i$ contributes nonzero residue precisely when
  • $E_i$ is strict (a component of the strict transform) and $E_i(\R)\neq \emptyset$, with $s_0 = -1/N_i$, or
  • $E_i$ is exceptional and meets at least three other components (real intersections counted once, complex twice).

There is no cancellation among exceptional divisor contributions in the unsigned $Z_{\mathrm{top},0}$. For the signed $Z_{\mathrm{top},0}^{\pm}$, cancellations may occur, but a pole persists whenever exactly one contributing divisor exists for a given value. In general, the set of poles for the signed variant is a subset of those for the unsigned.

5. Examples: The Real Cusp and Computation of Poles

Consider $f(x,y) = y^2 - x^3$. After three successive real blow-ups, the exceptional divisors and strict transform have the diagram:

$$E_1(2,2) — E_2(3,3) — E_3(5,6) — E_4(1,1) \;\;\;\backslash\;\;\;\;\;\quad\backslash \;\;\;\;\;E_5(1,1)$$

With the two-fold sum formula:

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

Partial fraction decomposition yields:

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

Thus, poles occur at $s=-1$ (from the strict transform with $N_4=1$) and $s=-5/6$ (from the exceptional divisor $E_3(5,6)$ with three-fold intersection) (Jaudon, 5 Jan 2026).

• Signed cases similarly yield poles at $s=-1$ and $s=-5/6$ (with different residues).

6. Relationship to the Complex and $b$-Function Conjectures

The Denef–Loeser philosophy for complex polynomials associates the poles of $Z_{\mathrm{top},x}(f,s)$ to roots of the local Bernstein–Sato polynomial $b_{f,x}(s)$. In complex settings, for reduced three-variable hyperplane arrangements, the formula for $Z_{\mathrm{top},0}(f,s)$ is:

$$Z_{\mathrm{top},0}(f,s) = \frac{\chi(\P^2 \setminus Z)}{ds+3} + \frac{\chi(Z \setminus Z_\mathrm{sing})}{s+1} + \sum_{m\geq2} U_m \frac{2-m + ms + 1}{ms + 2}$$

where $U_m$ counts singular points of multiplicity $m$.

Pole candidates arise at $s = -3/d$ (possible double pole if $d/3\in \Z$ and $U_{d/3}>0$), and $s=-2/m$ whenever $U_m > 0$ (Budur et al., 2010). Denef–Loeser conjecture ([A]$_{\mathrm{top}}$) asserts all poles are roots of $b_{f,x}(s)$. Budur–Saito–Yuzvinsky prove this conjecture for broad classes of hyperplane arrangements, including all reduced arrangements in $\C^3$.

Explicit calculation in complex and real settings showcase both similarities and subtle differences in pole-detection and cancellation phenomena.

7. Applications and Further Significance

Real motivic zeta functions and their specializations serve as invariants under blow-Nash equivalence, facilitating classification of real singularities from the arc space perspective (Jaudon, 5 Jan 2026). The explicit characterization of poles informs both computational approaches and theoretical developments in singularity theory, especially for understanding monodromy, $b$-functions, and motivic integration on arc spaces.

Sample computations, such as for hyperplane arrangements and plane curve singularities, illustrate sharpness of pole detection, occurrence of cancellations, and the influence of the real structure, with signed zeta functions encoding additional data on the signs of defining equations. These results demonstrate the power of motivic and topological invariants in distinguishing between subtle singularity types in the real algebraic context.