# WEIGHTED CAUCHY FORMULA AND THE POLYDISK NULLSTELLENSATZ

Mamadou Mboup, Alain Yger

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Mamadou Mboup, Alain Yger. WEIGHTED CAUCHY FORMULA AND THE POLYDISK NULL-STELLENSATZ. 9th IFAC Symposium on System Structure and Control (SSSC), 19th IFAC Workshop on Time Delay Systems (TDS), and 2nd IFAC Workshop on Control of Complex Systems (COSY), IFAC, Jun 2025, Gif-sur-Yvette, France. hal-05243845

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Distributed under a Creative Commons Attribution 4.0 International License
WEIGHTED CAUCHY FORMULA AND THE POLYDISK NULLSTELLENSATZ

MAMADOU MBOUP AND ALAIN YGER

ABSTRACT. We solve explicitly the polydisk nullstellensatz in the integral domain of structural stable $n$-dimensional systems and comment about the related polydisk membership problem.

1. INTRODUCTION

The motivation of this study comes from control theory where it has been shown in Bouzidi-Quadrat-Rouillier (2019); Cluzeau-Moroz-Quadrat (2024), that the problem of internal stabilization of an $nD$ system can be stated as the polydisk nullstellensatz problem.

We give an explicit solution of this problem in Theorem 1 below, along with explicit formula for the associated cofactors.

We start with the simple 1D case to fix notations.

Let $\overline{\mathbb{D}}$ be the unit closed disk in the complex plane and $p_1, \dots, p_m$ be $m$ univariate polynomials in $\mathbb{R}[X]$ such that $\{\zeta \in \mathbb{C} : p_j(\zeta) = 0, j = 1, \dots, m\} \cap \overline{\mathbb{D}}_r = \emptyset$ for some $r > 1$, where $\overline{\mathbb{D}}_r := r \overline{\mathbb{D}}$.
We denote the $m$ vector of complex valued functions $(p_1, \dots, p_m)$ by $\mathbf{p}$ and let $\|\mathbf{p}\|$ be its Euclidean norm. Let also $\gamma_r$ be the real analytic 1-cycle $\theta \in \mathbb{R}/(2\pi\mathbb{Z}) \hookrightarrow r e^{i\theta}$ and $\partial/\partial\bar{\zeta} := (\partial/\partial\zeta + i\partial/\partial\eta)/2$ be the Cauchy-Riemann operator acting on smooth functions of $\zeta = \zeta + i\eta$.

Then the following identity (between functions of z) holds in $\overline{\mathbb{D}}$:

$$ (1) \quad 1 = \sum_{\kappa=0}^{\infty} \left( \frac{1}{2\pi} \int_{0}^{2\pi} \left( \sum_{j=1}^{m} p_j(z) \frac{\overline{p_j(re^{i\theta})}}{\|\mathbf{p}(re^{i\theta})\|^2} \right)^2 e^{-i\kappa\theta} d\theta \right) \left(\frac{z}{r}\right)^{\kappa} \\ \qquad + \frac{2}{\pi} \sum_{j=1}^{m} p_j(z) \left( \int_{\overline{\mathbb{D}}_r} \frac{\overline{p_j(\zeta)}}{\|\mathbf{p}(\zeta)\|^2} \frac{\partial}{\partial \bar{\zeta}} \left( \sum_{\ell=1}^{m} \frac{\overline{p_\ell(\zeta)}}{\|\mathbf{p}(\zeta)\|^2} \frac{p_\ell(\zeta) - p_\ell(z)}{\zeta - z} \right) d\zeta d\eta \right). $$

This follows from the Cauchy-Pompeiu integral representation formula at $z \in \overline{\mathbb{D}} \subset \mathbb{D}_r$

$$ 1 = \frac{1}{2i\pi} \left( \int_{\gamma_r} \frac{f_z(\zeta)}{\zeta - z} d\zeta - \int_{\overline{\mathbb{D}}_r} \frac{\bar{\partial}(f_z(\zeta) d\zeta)}{\zeta - z} \right), $$

where

$$ \begin{aligned} \bar{\partial}(\omega(\zeta) d\zeta) &= \bar{\partial}(\omega(u + iv) (du + idv)) = \frac{\partial \omega}{\partial \bar{\zeta}}(\zeta) d\bar{\zeta} \wedge d\zeta \\ &= i \left( \frac{\partial}{\partial u} + i \frac{\partial}{\partial v} \right) (\omega(u + iv)) du \wedge dv, \end{aligned} $$

<footnote>Date: June 25, 2025.</footnote>

<footnote n="1991">1991 Mathematics Subject Classification. Primary 93B25, 93C05, 93C35; Secondary 32A08, 32A10.</footnote>

<footnote>Key words and phrases. Stability; Algebraic systems theory; Multidimensional systems; Polynomial with several complex variables.</footnote>
see for example (Berenstein-Gay, 1991, Proposition 2.1.2), for the smooth function

$$ (2) \qquad f_z : \zeta \in \bar{\mathbb{D}}_r \mapsto \left( \sum_{j=1}^{m} p_j(z) \frac{\overline{p_j(\zeta)}}{\| \mathbf{p}(\zeta) \|^2} \right)^2. $$

Observe that for each $j = 1, \dots, m$, the coefficients of the polynomial in $X$

$$ u_j(X) = \int_{\bar{\mathbb{D}}_r} \frac{\overline{p_j(\zeta)}}{\|\mathbf{p}(\zeta)\|^2} \frac{\partial}{\partial \bar{\zeta}} \left( \sum_{\ell=1}^{m} \frac{\overline{p_j(\zeta)}}{\|\mathbf{p}(\zeta)\|^2} \frac{p_\ell(\zeta) - p_\ell(X)}{\zeta - X} \right) d\zeta d\eta $$

are linear combinations with real coefficients of integrals

$$ \int_0^r \left( \frac{1}{2\pi} \int_0^{2\pi} \frac{e^{i(\mu-\nu)\theta}}{\|\mathbf{p}(\rho e^{i\theta})\|^2} d\theta \right) \rho^{\mu+\nu+1} d\rho, $$

where $\mu, \nu \in \mathbb{N}$ and $\gamma \in \mathbb{N}^*$. All such integrals are real since Fourier coefficients of real and even $2\pi$ periodic functions are. Let $c_r(\mathbf{p})$ be the strictly positive constant

$$ c_r(\mathbf{p}) = \left( \frac{1}{2\pi} \int_0^{2\pi} \|\mathbf{p}(re^{i\theta})\|^{-2} d\theta \right) \max_{\bar{\mathbb{D}}} \|p\|^2. $$

Given $\epsilon \in ]0, 1[$ and $K_\epsilon \in \mathbb{N}$ such that

$$ r^{K_\epsilon} \geq \frac{c_r(\mathbf{p})}{r-1} \frac{1}{\epsilon}, $$

it follows from (1) that the polynomial function (also with real coefficients for the same reason as above)

$$ q^{[\epsilon]}(z) = \sum_{\kappa=0}^{K_\epsilon} \left( \frac{1}{2\pi} \int_0^{2\pi} \left( \sum_{j=1}^{m} p_j(z) \frac{\overline{p_j(re^{i\theta})}}{\|\mathbf{p}(re^{i\theta})\|^2} \right)^2 e^{-i\kappa\theta} d\theta \right) \left(\frac{z}{r}\right)^{\kappa} + \frac{2}{\pi} \sum_{j=1}^{m} p_j(z) u_j(z) $$

is such that $1 - \epsilon \le |q^{[\epsilon]}| \le 1 + \epsilon$ in $\bar{\mathbb{D}}$. Then, the polynomial identity

$$ q^{[\epsilon]}(X) = \sum_{\kappa=0}^{K_\epsilon} \left( \frac{1}{2\pi} \int_0^{2\pi} \left( \sum_{j=1}^{m} p_j(X) \frac{\overline{p_j(re^{i\theta})}}{\|\mathbf{p}(re^{i\theta})\|^2} \right)^2 e^{-i\kappa\theta} d\theta \right) \left(\frac{X}{r}\right)^{\kappa} + \frac{2}{\pi} \sum_{j=1}^{m} p_j(X) u_j(X) $$

is a realization in $\mathbb{R}[X]$ of the disk nullstellensatz (according to the terminology introduced in Bridges and al. (2003) within the complex instead of real setting) at the precision at most $\epsilon \in (0, 1)$.

The goal of the present short note is to transpose such observation from the univariate setting to the multivariate setting, which is done in §2, see Theorem 1. This makes effective in $\mathbb{R}[X_1, \dots, X_n]$ the positive answer to the polydisk nullstellensatz in $\mathbb{C}[X_1, \dots, X_n]$, such as proposed in Bridges and al. (2003). Our result relies on the powerful machinery which consists in weighting Bochner-Martinelli formula in order to transform such integral representation formula into a division formula in $\mathbb{R}[X_1, \dots, X_n]$. In §3, we also comment about this result in order to address the so-called effective polydisk membership.

## 2. THE POLYDISK NULLSTELLENSATZ

Let $\bar{\mathbb{D}}^n$ be the closed unit polydisk in $\mathbb{C}^n$ and $p_1, \dots, p_m$ be $m$ polynomials in $n$ variables with real coefficients which satisfy $\{\zeta \in \mathbb{C}^n : p_j(\zeta) = 0, j = 1, \dots, m\} \cap \bar{\mathbb{D}}^n_r = \emptyset$ for some $r_1, \dots, r_n > 1$,
where $\overline{\mathbb{D}}_r^n := \overline{\mathbb{D}}_{r_1} \times \cdots \times \overline{\mathbb{D}}_{r_n}$. We attach to $\mathbf{p} = (p_1, \dots, p_m)$ a matrix $[h_{j,k}]_{1 \le j \le m, 1 \le k \le n}$ of elements in $\mathbb{R}[X_1, \dots, X_n, W_1, \dots, W_n]$ with, for $j = 1, \dots, m$,

$$p_j(W) - p_j(X) = \sum_{k=1}^{n} (W_k - X_k) h_{j,k}(X, W).$$

Observe that such construction is not unique. A possible one is induced, for $\ell = 1, \dots, n$, by the recursive process:

$$ \begin{aligned} p_j(W) &= (W_1 - X_1) h_{j,1}(X_1, W) + p_j(X_1, W_2, \dots, W_n) \\ p_j(X_1, \dots, X_{\ell-1}, W_\ell, \dots, W_n) &= (W_\ell - X_\ell) h_{j,\ell}(X_1, \dots, X_\ell, W) \\ &\quad + p_j(X_1, \dots, X_\ell, W_{\ell+1}, \dots, W_n). \end{aligned} $$

Consider $\mathbf{k} : 1 \le k_1 < \cdots < k_{n-\ell} \le n$ a non-empty ordered subset of $\{1, \dots, n\}$ with cardinal $n-\ell$ between 1 and $n$ (note that we omit to specify the dependence of $\mathbf{k}$ in its co-length $\ell = n - \text{card}(\mathbf{k})$ in order to simplify notations). Let

$$ \bullet \gamma_{r,k} : \theta \in (\mathbb{R}/(2\pi\mathbb{Z}))^{n-\ell} \mapsto (r_{k_1} e^{i\theta_1}, \dots, r_{k_{n-\ell}} e^{i\theta_{n-\ell}}) \in \mathbb{C}^{n-\ell}, $$

$$ \bullet \bar{\mathbb{D}}_{r,k}^{\ell} = \bar{\mathbb{D}}_{r_{k'_1}} \times \cdots \times \bar{\mathbb{D}}_{r_{k'_\ell}} \text{ when } \ell > 0 \text{ with } \{1, \dots, n\} \setminus \mathbf{k} : 1 \le k'_1 < \cdots < k'_\ell \le n, $$

$$ \bullet \bar{\mathbb{D}}_{r,k}^0 = \{1\}. $$

For each $\omega \in \gamma_{r,k}((\mathbb{R}/(2\pi\mathbb{Z}))^{n-\ell})$ and for each $j = 1, \dots, n$, we define the polynomial $p_{j;k,\omega} \in C[Y_1, \dots, Y_\ell]$ as follows: in $p_j(X_1, \dots, X_n)$, one replaces each variable $X_{k_\mu}$ with $\mu = 1, \dots, n-\ell$ by $\omega_\mu$ (unless $\ell = 0$) and each variable $X_{k'_\nu}$ for $\nu = 1, \dots, \ell$ by $Y_\nu$. In case $\ell = 0$ and $\omega = (r_1 e^{i\theta_1}, \dots, r_n e^{i\theta_n})$, $p_{j,k,\omega}$ is then the constant $p_j(\omega)$. Let also $\mathbf{p}_{k,\omega} := (p_{1;k,\omega}, \dots, p_{m;k,\omega})$. In the same vein, let each polynomial $h_{j,k;\omega}(X, Y)$ ($1 \le j \le m, 1 \le k \le n$) be defined as follows: one replaces in $h_{j,k}(X, W)$ each variable $W_{k_\mu}$ with $\mu$ between 1 and $n-\ell$ by $\omega_\mu$ (unless $\ell = 0$) and each variable $W_{k'_\nu}$ with $\nu$ between 1 and $\ell$, where $\{1, \dots, n\} \setminus \mathbf{k} : 1 \le k'_1 < \cdots < k'_\ell \le n$, by $Y_\nu$. We reserve for the time being such $\mathbf{p}_{k,\omega}$, $h_{j,k;\omega}$ ($1 \le j \le m, 1 \le k \le n$) until the formulation of forthcoming Proposition 2.

Attached to $\mathbf{p}$ and the $h_{j,k}$'s ($1 \le j \le m, 1 \le k \le n$) is the smooth $(1,0)$ form in $\overline{\mathbb{D}}_r^n$ which is parametrized holomorphically by $z \in \overline{\mathbb{D}}^n$ as

$$ (3) \qquad Q_z^0 : \zeta \in \overline{\mathbb{D}}_r^n \mapsto \sum_{j=1}^{m} \frac{\overline{p_j(\zeta)}}{\|\mathbf{p}(\zeta)\|^2} \sum_{k=1}^{n} h_{j,k}(z, \zeta) \, d\zeta_k. $$

Attached to $\mathbf{r} = (r_1, \dots, r_n)$ is a collection of $n$ smooth $(1,0)$ forms in $\overline{\mathbb{D}}_r^n$, each of them being parametrized holomorphically by $z \in \overline{\mathbb{D}}^n$ as

$$ Q_z^k : \zeta \in \overline{\mathbb{D}}_r^n \mapsto \frac{\bar{\zeta}_k}{r_k^2 - \bar{\zeta}_k z_k} \, d\bar{\zeta}_k $$

for $k = 1, \dots, n$. For any $k = 0, \dots, n$, one introduces now the smooth function in $\overline{\mathbb{D}}_r^n$ which is parametrized holomorphically by $z \in \overline{\mathbb{D}}^n$ as $\phi_z^k$:

$$ \phi_z^k : \zeta \in \overline{\mathbb{D}}_r^n \mapsto 1 + \left( \sum_{\kappa=1}^{n} (z_\kappa - \zeta_\kappa) \frac{\partial}{\partial \zeta_\kappa} \right) Q_z^k(\zeta), $$
where $\rfloor$ is the interior left multiplication of (1,0) vector fields with (1,0) differential forms, namely $(\partial/\partial\zeta_\kappa)\rfloor d\zeta_\kappa'$ either equals 0 whenever $\kappa \neq \kappa'$ or 1 when $\kappa = \kappa'$. One has then for $z \in \bar{\mathbb{D}}^n$ and $\zeta \in \bar{\mathbb{D}}_r^n$

$$ (4) \qquad \phi_z^0(\zeta) = \sum_{j=1}^{m} \frac{\overline{p_j(\zeta)}}{\|\boldsymbol{p}(\zeta)\|^2} p_j(z), $$

while

$$ (5) \qquad \phi_z^k(\zeta) = \frac{r_k^2 - |\zeta_k|^2}{r_k^2 - \bar{\zeta}_k z_k} $$

for $k = 1, \dots, n$. It is important for the sequel to point out that for any $k = 1, \dots, n, z \in \bar{\mathbb{D}}^n$, $\zeta \in \bar{\mathbb{D}}_r^n$,

$$ (6) \qquad \mathrm{Re} (\phi_z^k(\zeta)) = \frac{r_k^2 - |\zeta_k|^2}{|r_k^2 - \bar{\zeta}_k z_k|^2} (r_k^2 - \mathrm{Re} (\zeta_k \bar{z}_k)) > 0 $$

$$ |\phi_z^k(\zeta)| = (r_k + |\zeta_k|) \frac{r_k - |\zeta_k|}{|r_k^2 - \bar{\zeta}_k z_k|} \le 2. $$

Instead of $f_z$ as in (2), one introduces first (following Andersson (1992) or (Adachi and al., 1999, §2)) a strictly positive parameter $\lambda$ and the holomorphic function $\Gamma$ of $2n + 1$ complex variables $\omega_0, \dots, \omega_{2n}$ in

$$ \{\omega \in \mathbb{C}^{2n+1} : \mathrm{Re} \, \omega_k > 0 \text{ for } k = 1, \dots, 2n\} $$

defined by

$$ \Gamma(\omega) = \omega_0^{n+1} \left( \prod_{k=1}^{2n} \omega_k \right)^{\lambda}. $$

For any $\kappa \in \mathbb{N}^{2n+1}$ with $|\kappa| = \kappa_0 + \dots + \kappa_{2n} \le n$, the smooth function $f_{\lambda, \kappa, z}$ in $\bar{\mathbb{D}}_r^n$ parametrized by $\lambda, \kappa$ and holomorphically by $z \in \bar{\mathbb{D}}^n$ is defined by

$$ f_{\lambda, \kappa, z}(\zeta) = \left( \prod_{\nu=0}^{2n} \left( \frac{1}{\kappa_\nu!} \left( \frac{\partial}{\partial \omega_\nu} \right)^{\kappa_\nu} \right) (\Gamma) \right) (\phi_z^0(\zeta), \Phi_z(\zeta), \Phi_z(\bar{\zeta})), $$

where $\Phi_z(\zeta) = (\phi_z^1(\zeta), \dots, \phi_z^n(\zeta))$. Note that such definition is licit in view of the first assertion in (6). With such preliminary material within hands, one may represent in $\bar{\mathbb{D}}^n$ the constant function $z \in \bar{\mathbb{D}}^n \mapsto 1$ through an explicit integral representation formula which is an intermediate step towards a multivariate analog of (1).

**Proposition 1.** For any $z \in \bar{\mathbb{D}}^n$ and any $\lambda > 0$, one has

$$ (7) \qquad 1 = \frac{1}{(2i\pi)^n} \sum_{\{\kappa \in \mathbb{N}^{2n+1} : |\kappa| = n\}} \int_{\bar{\mathbb{D}}_r^n} f_{\lambda, \kappa, z}(\zeta) (\bar{\partial} Q_z^0(\zeta))^{\kappa_0} \bigwedge_{k=1}^n (\bar{\partial} Q_z^k(\zeta))^{\kappa_k + \kappa_{k+1}}. $$

*Proof of Proposition 1.* The assertion is a particular case of (Vidras-Yger, 2023, Proposition 4.34, formula (4.84)). To be more precise, let $\mathbb{E}$ be the $n$-holomorphic bundle over $\mathbb{C}^n \times \mathbb{C}^n$ defined as the pull-back through the projection $(z, \zeta) \mapsto \zeta$ of the holomorphic tangent bundle over $\mathbb{C}^n$. Its holomorphic section

$$ (z, \zeta) \mapsto \Delta(z, \zeta) = \sum_{\kappa=1}^{n} (\zeta_{\kappa} - z_{\kappa}) \frac{\partial}{\partial \zeta_{\kappa}} \in \mathbb{E}_{z, \zeta} $$
is such that

$$ \Delta^{-1}(\{0\}) = \{(z, \zeta) \in \mathbb{C}^n \times \mathbb{C}^n : \zeta = z\} \subset \{(z, \zeta) \in \mathbb{C}^n \times \mathbb{C}^n : d\Delta(z, \zeta) \neq 0\} $$

and the sheaf of sections associated to the Koszul complex ($K_\bullet^\Delta$) built over $\mathbb{C}^n \times \mathbb{C}^n$ from the interior multiplication $\rfloor_\Delta : \mathbb{E}^* \to \mathbb{C}$ is locally exact except at the 0 level since one has codim $\Delta^{-1}(\{0\}) = n = \text{rank } \mathbb{E}$. Proposition 1 relies on the fact that

$$ \mathcal{W}_\lambda : (z, \zeta) \in \overline{\mathbb{D}}^n \times \overline{\mathbb{D}}_r^n \mapsto \sum_{\{\kappa \in \mathbb{N}^{2n+1} : |\kappa| \le n\}} f_{\lambda, \kappa, z}(\zeta) \left( \frac{\partial Q_z^0(\zeta)}{2i\pi} \right)^{\kappa_0} \bigwedge_{k=1}^n \left( \frac{\partial Q_z^k(\zeta)}{2i\pi} \right)^{\kappa_k + \kappa_{k+n}} $$

extends to some open neighborhood $\mathcal{U}$ of $\overline{\mathbb{D}}^n \times \overline{\mathbb{D}}_r^n$ in $\mathbb{C}^n \times \mathbb{C}^n$ as a (smooth) global Bochner-Martinelli C-valued weight in $\mathcal{U}$ (denoted also by $\mathcal{W}_\lambda$) with respect to the Koszul complex ($K_\bullet^\Delta$). Namely $\mathcal{W}_\lambda = \sum_{\chi \in \mathbb{N}} \mathcal{W}_\lambda^\chi$, where $\mathcal{W}_\lambda^\chi$ is a smooth $\wedge^\chi \mathbb{E}^*$-valued differential form with bidegrees $(0,0)$ in $z$ and $(0,\chi)$ in $\zeta$, holomorphic in $z$, and such that

$$ (8) \qquad (2i\pi)\rfloor_\Delta - \bar{\partial}_\zeta) \mathcal{W}_\lambda(z, \zeta) = 0 \text{ in } \mathcal{U}, \quad \mathcal{W}_\lambda^0(z, z) = 1. $$

Both properties (8) are immediate to check. A strategy to generate and combine smooth C-valued global Bochner-Martinelli weights was elaborated in the pioneering works Andersson-Berndtsson (1982) and Andersson-Passare (1988), see also (Berenstein and al., 1993, §2.3). We refer the reader to (Vidras-Yger, 2023, Proposition 4.20) for a presentation of such construction of C-valued smooth Bochner-Martinelli weights with respect to the Koszul complex ($K_\bullet^\Delta$), from which $\mathcal{W}_\lambda$ is precisely the archetypal product. One may then apply (Vidras-Yger, 2023, Proposition 4.34) with $\Omega = \overline{\mathbb{D}}_r^n$, $\Omega' = (1+\varepsilon)\overline{\mathbb{D}}^n$ with $0 < \varepsilon \ll 1$ in order that $\overline{\mathbb{D}}^n \subset \overline{\Omega'} \subset \subset \Omega$, $E = \mathbb{C}$, $U \subset \mathbb{C}^n \setminus \overline{\Omega'}$ a neighborhood of $\partial\Omega$, together with a smooth form

$$ (z, \zeta) \in \Omega' \times (\overline{\Omega} \cap U) \mapsto \sigma_{\Omega', U}(z, \zeta) = \sum_{k=1}^{n} \sigma_{\Omega', U, k}(z, \zeta) d\zeta_k $$

which is realized here through a partition of unity in order as to depend holomorphically of $z \in \Omega'$ and satisfy $\rfloor_\Delta \sigma_{\Omega', U}(z, \zeta) \equiv 1$ in $\overline{\Omega}' \times (\overline{\Omega} \cap U)$. The choice of $U$ and $\sigma_{\Omega', U}$ are here irrelevant since $\phi_z^1 \cdots \phi_z^n$ vanishes on $\partial \overline{\mathbb{D}}_r^n$ for any $z \in \overline{\mathbb{D}}^n$, see (5). So does $\mathcal{W}_\lambda(z, \cdot)$ since the multi-indexes $\kappa$ such that $\kappa_j + \kappa_{j+n} = 0$ or $1$ are the only ones to be taken in consideration in the expression of $\mathcal{W}_\lambda$ and the product $\phi_z^1 \cdots \phi_z^n$ remains therefore present in each corresponding $f_{\lambda, \kappa, z}$. The assertion follows from (Vidras-Yger, 2023, Proposition 4.34, (4.84)) with $s = 1$. $\square$

**Proposition 2.** For any $z \in \overline{\mathbb{D}}^n$, one has

$$ (9) \quad 1 = \sum_{\ell=0}^{n-1} \binom{n+1}{n+1-\ell} \sum_{1 \le k_1 < \dots < k_{n-\ell} \le n} \left( \frac{1}{(2i\pi)^{n-\ell}} \int_{\gamma_{r,k}} \Omega_{z,k}(\omega) \bigwedge_{\mu=1}^{n-\ell} \frac{d\omega_\mu}{\omega_\mu - z_{k_\mu}} \right) + (n+1) \sum_{j=1}^{m} p_j(z) u_j(z), $$

where, for $1 \le j \le m$,

$$ (10) \qquad (2i\pi)^n u_j(z) = \int_{\overline{\mathbb{D}}_r^n} \frac{\overline{p_j(\zeta)}}{\|\mathbf{p}(\zeta)\|^2} \left( \bar{\partial}_\zeta \left( \sum_{j'=1}^{m} \frac{\overline{p_{j'}(\zeta)}}{\|\mathbf{p}(\zeta)\|^2} \sum_{k=1}^{n} h_{j',k}(z, \zeta) d\zeta_k \right) \right)^n $$
and for each $\mathbf{k} : 1 \le k_1 < \dots < k_{n-\ell} \le n$ with $\ell = 0, \dots, n-1$, for each $\omega \in \gamma_{\mathbf{r},\mathbf{k}}((\mathbb{R}/(2\pi\mathbb{Z}))^{n-\ell})$, (11)

$$ (2i\pi)^{\ell} \Omega_{z,k}(\omega) = \int_{\overline{\mathbb{D}}_{r,k}^{\ell}} \left( \sum_{j=1}^{m} \frac{\overline{p_{j;k,\omega}(\zeta)}}{\|\mathbf{p}_{k,\omega}(\zeta)\|^2} p_j(z) \right)^{n+1-\ell} \left( \bar{\partial}_{\zeta} \left( \sum_{j'=1}^{m} \frac{\overline{p_{j';k,\omega}(\zeta)}}{\|\mathbf{p}_{k,\omega}(\zeta)\|^2} \sum_{k'=1}^{n} h_{j',k;k,\omega}(z,\zeta) d\zeta_{k'} \right) \right)^{\ell} $$

$$ (12) \qquad = \sum_{j=1}^{m} \Omega_{z,k,j}(\omega) p_j(z). $$

*Proof of Proposition 2.* For any $\mathbf{\kappa} \in \mathbb{N}^{2n+1}$ such that $|\mathbf{\kappa}| = n$, consider the (possibly empty) subset of $\{1, \dots, n\}$ defined as

$$ E(\mathbf{\kappa}) := \{k \in \{1, \dots, n\} : \kappa_k + \kappa_{k+n} = 1\}. $$

When such subset is empty, that is $\mathbf{\kappa} = (n, 0, \dots, 0)$, one has for $\lambda > 0$, $z \in \overline{\mathbb{D}}^n$ and $\zeta \in \overline{\mathbb{D}}_r^n$ that

$$ (13) \qquad f_{\lambda, \kappa, z}(\zeta) = (n+1) \phi_z^0(\zeta) \left( \prod_{k=1}^{n} \phi_z^k(\zeta) \right)^{2\lambda} $$

$$ (14) \qquad = (n+1) \left( \sum_{j=1}^{m} \frac{\overline{p_j(\zeta)}}{\|\mathbf{p}(\zeta)\|^2} p_j(z) \right) \left( \prod_{k=1}^{n} \frac{r_k^2 - |\zeta_k|^2}{r_k^2 - \bar{\zeta}_k z_k} \right)^{2\lambda}. $$

Suppose now that $\mathrm{card}(E(\mathbf{\kappa})) = n - \ell$ with $0 \le \ell \le n - 1$. Then

$$ (15) \qquad f_{\lambda, \kappa, z}(\zeta) = \frac{1}{2^{n-\ell}} \binom{n+1}{n+1-\ell} \left( \sum_{j=1}^{m} \frac{\overline{p_j(\zeta)}}{\|\mathbf{p}(\zeta)\|^2} p_j(z) \right)^{n+1-\ell} \left( \prod_{k \notin E(\mathbf{\kappa})} \frac{r_k^2 - |\zeta_k|^2}{r_k^2 - \bar{\zeta}_k z_k} \right)^{2\lambda} \prod_{k \in E(\mathbf{\kappa})} \left( 2\lambda \left( \frac{r_k^2 - |\zeta_k|^2}{r_k^2 - \bar{\zeta}_k z_k} \right)^{2\lambda-1} \right). $$

Observe that for any $k \in E(\mathbf{\kappa})$,

$$ \begin{aligned} 2\lambda \left( \frac{r_k^2 - |\zeta_k|^2}{r_k^2 - \bar{\zeta}_k z_k} \right)^{2\lambda-1} (\bar{\partial}_{\zeta_k} Q_z^k)(\zeta) &= 2\lambda \left( \frac{r_k^2 - |\zeta_k|^2}{r_k^2 - \bar{\zeta}_k z_k} \right)^{2\lambda-1} \frac{r_k^2}{(r_k^2 - \bar{\zeta}_k z_k)^2} d\bar{\zeta}_k \wedge d\zeta_k \\ &= -\frac{1}{\zeta_k} \bar{\partial}_{\zeta_k} \left( 1 - \frac{|\zeta_k|^2}{r_k^2} \right)^{2\lambda} \wedge \frac{d\zeta_k}{(1 - \bar{\zeta}_k z_k / r_k^2)^{2\lambda+1}}. \end{aligned} $$

We now let $\lambda$ tend to $0_+$ in (7). According to the second assertion in (6), one has that for any $k \in \{1, \dots, n\}$, for any $z \in \overline{\mathbb{D}}^n$ and almost each $\zeta_k \in \overline{\mathbb{D}}_{r_k}$,

$$ \lim_{\lambda \to 0^+} \left( \frac{r_k^2 - |\zeta_k|^2}{r_k^2 - \bar{\zeta}_k z_k} \right)^{2\lambda} = 1, $$

such convergence being dominated. On the other hand, for any $\mathbf{\kappa}$ such that $E(\mathbf{\kappa}) \neq \emptyset$ with cardinal $n - \ell$ with $0 \le \ell \le n - 1$ and any $k \in E(\mathbf{\kappa})$, one has

$$ \lim_{\lambda \to 0^+} \frac{d\zeta_k}{(1 - \bar{\zeta}_k z_k / r_k^2)^{2\lambda+1}} = \frac{r_k^2}{r_k^2 - \bar{\zeta}_k z_k} d\zeta_k $$
in the space of (1,0) smooth forms in $\overline{\mathbb{D}}^n \times \overline{\mathbb{D}}_r^n$, while for any smooth form $(z, \zeta) \mapsto \varphi_z(\zeta)$ in $\overline{\mathbb{D}}^n \times \overline{\mathbb{D}}_r^n$ with bidegrees (0,0) in $z$ and $(\ell, \ell)$ in $\zeta$

$$ \lim_{\lambda \to 0^+} \int_{\overline{\mathbb{D}}_r^n} \bigwedge_{k \in E(\kappa)} \left( -\bar{\partial}_{\zeta_k} \left( 1 - \frac{|\zeta_k|^2}{r_k^2} \right)^{2\lambda} \wedge d\zeta_k \right) \wedge \varphi_z(\zeta) = \int_{\gamma_{r,k}} \varphi_z(\zeta) $$

according to the jump formula. Taking into account the estimates (6) which secure the Lebesgue domination theorem and the expressions (14) or (15) for the $f_{\lambda,k,z}$ when $\lambda > 0$ and $z \in \overline{\mathbb{D}}^n$, formula (9) is asymptotically a consequence of (7) since, for $k: 1 \le k_1 < \dots < k_{n-\ell} \le n$ with $0 \le \ell \le n-1$, there are $2^{n-\ell}$ contributions

$$ \lim_{\lambda \to 0^+} \int_{\overline{\mathbb{D}}_r^n} f_{\lambda, \kappa, z}(\zeta) (\bar{\partial} Q_z^0(\zeta))^{\kappa_0} \bigwedge_{k=1}^n (\bar{\partial} Q_z^k(\zeta))^{\kappa_k + \kappa_{k+n}} $$

with $\kappa_k + \kappa_{k+n} = 1$ if and only if $k \in \{k_1, \dots, k_{n-\ell}\}$, which compensates the factor $1/2^\ell$ in front of the right-hand side of (15). $\square$

**Theorem 1 (polydisk nullstellensatz).** Let $r = \min_k r_k$ and $c_r(p)$ be the maximum of all positive numbers

$$ \frac{1}{(2\pi)^{n-\ell}} \int_{(\mathbb{R}/(2\pi\mathbb{Z}))^{n-\ell}} |\Omega_{z,k}(r_{k_1}e^{i\theta_1}, \dots, r_{k_{n-\ell}}e^{i\theta_{n-\ell}})| d\theta $$

for $\ell = 0, \dots, n-1$, $1 \le k_1 < \dots < k_{n-\ell} \le n$ and $z \in \overline{\mathbb{D}}^n$. Then, for $0 < \epsilon < 1$ and $K_\epsilon \in \mathbb{N}$ such that

$$ (16) \qquad r^{K_\epsilon+1-n} \ge n(2^n-1) \frac{c_r(p)}{(r-1)^n} \frac{1}{\epsilon}, $$

the polynomial

$$ \begin{aligned} (17) \quad q^{[\epsilon]}(X) = & \sum_{\ell=0}^{n-1} \frac{1}{(2i\pi)^{n-\ell}} \binom{n+1}{n+1-\ell} \sum_{j=1}^{m} p_j(X) \left( \sum_{1 \le k_1 < \dots < k_{n-\ell} \le n} \sum_{\{\kappa \in \mathbb{N}^{n-\ell}: \max \kappa_\mu \le K_\epsilon\}} \right. \\ & \left. \left( \int_{\gamma_{r,k}} \frac{\Omega_{X,k,j}(\omega)}{\omega_1^{\kappa_1} \dots \omega_{n-\ell}^{\kappa_{n-\ell}}} \bigwedge_{\mu=1}^{n-\ell} \frac{d\omega_\mu}{\omega_\mu} \right) \prod_{\mu=1}^{n-\ell} \left( \frac{X_{k_\mu}}{r_{k_\mu}} \right)^{\kappa_\mu} \right. \\ & \left. + (n+1) \sum_{j=1}^{m} p_j(X) u_j(X) \right) \end{aligned} $$

with $u_1, \dots, u_m$ as in (10) and the $\Omega_{X,k,j}$ as in (11) with $X$ instead of $z$ belongs to the ideal generated by the $p_j$'s in $\mathbb{R}[X_1, \dots, X_n]$ and satisfies $1 - \epsilon \le |q^{[\epsilon]}| \le 1 + \epsilon$ in $\overline{\mathbb{D}}^n$, hence solves in $\mathbb{R}[X_1, \dots, X_n]$ the polydisk nullstellensatz.

*Proof of Theorem 1.* For any $\ell = 0, \dots, n-1$ and

$$ k: 1 \le k_1 < \dots < k_{n-\ell} \le n, $$

one has for $z \in \overline{\mathbb{D}}^n$ that

$$ \int_{\gamma_{r,k}} \Omega_{z,k}(\omega) \bigwedge_{\mu=1}^{n-\ell} \frac{d\omega_\mu}{\omega_\mu - z_{k_\mu}} = \sum_{\kappa \in \mathbb{N}^{n-\ell}} \left( \int_{\gamma_{r,k}} \frac{\Omega_{z,k}(\omega)}{\omega_1^{\kappa_1} \cdots \omega_{n-\ell}^{\kappa_{n-\ell}}} \bigwedge_{\mu=1}^{n-\ell} \frac{d\omega_\mu}{\omega_\mu} \right) \prod_{\mu=1}^{n-\ell} \left( \frac{z_{k_\mu}}{r_{k_\mu}} \right)^{\kappa_\mu}. $$

Since

$$ \sum_{\{\kappa \in \mathbb{N}^{n-\ell}: \max_{\mu} \kappa_{\mu} > K_{\epsilon}\}} \prod_{\mu=1}^{n-\ell} \left(\frac{1}{r_{k_{\mu}}}\right)^{\kappa_{\mu}} \le \frac{n-\ell}{r-1} \frac{1}{r^{K_{\epsilon}}} \left(\frac{r}{r-1}\right)^{n-\ell-1} \le \frac{n}{(r-1)^n} \frac{1}{r^{K_{\epsilon}+1-n}}, $$
one has $1 - \epsilon \le |q^{[\epsilon]}| \le 1 + \epsilon$ in $\overline{\mathbb{D}}^n$ according to the identity (9). It is also clear from Proposition 2 that $q^{[\epsilon]}$ lies in the ideal generated by the $p_j$'s in $\mathbb{C}[X_1, \dots, X_n]$, see the explicit expressions (11). The fact that all polynomials implied as factors of monomial expressions in the $p_j$'s in the explicit expression of $q^{[\epsilon]}$ lie in $\mathbb{R}[X_1, \dots, X_n]$ follows as in the univariate setting from a Fourier analysis argument: any Fourier coefficient of a real and even $2\pi$-periodic function is real. $\square$

2.1. **Remark.** The proof of Theorem 1 relies of a division process in $\overline{\mathbb{D}}^n$ (applied here to the function $z \to 1$ with respect to the ideal generated by the polynomial functions $z \mapsto p_j(z)$ for $j = 1, \dots, m$) which is reminiscent to the Wiener filtering process : namely express 1 as

$$1 = \lim_{\eta \to 0^+} \sum_{j=1}^{m} \frac{\overline{p_j(z)}}{\|\mathbf{p}(z)\|^2 + \eta} p_j(z),$$

where $\eta > 0$ is interpreted as a ratio $\frac{\text{noise}}{\text{signal}}$. Such Wiener filtering process governs the weighting of Cauchy representation formula (in order to transform it into a division formula) which is constantly used in our approach, see e.g. the expressions of the algebraic expressions $Q_z^0$ and $\phi_z^0$ respectively in (3) and (4).

2.2. **Comments about the result and its implementation.** Given $r_1, \dots, r_n > 1$ with $r := \min_j r_j$ such that the polynomial entries $p_1, \dots, p_m \in \mathbb{R}[X]$ do not share any common zero in the closed polydisk $\overline{\mathbb{D}}^n$ and $0 < \epsilon < 1$, formula (17) produces explicit cofactors $u_1^{[\epsilon]}, \dots, u_m^{[\epsilon]} \in \mathbb{R}[X]$, each with total degree at most $n(d + K_\epsilon)$, where $d := \max_j \deg p_j$ and the choice of $K_\epsilon$ is subordinate to (16), such that $z \in \overline{\mathbb{D}}^n$ implies $|u^{[\epsilon]}(z)| \in [1 - \epsilon, 1 + \epsilon]$, where $q^{[\epsilon]} = \sum_{j=1}^m u_j^{[\epsilon]} p_j$. According to their explicit expressions in (17), see also (10) and (11), the coefficients of such $u_j^{[\epsilon]}$'s are integrals on spherical domains which can be numerically approximated with the fft machinery. If $\epsilon < 1/2$, a numerical tolerance of error

$$\text{tol}_{\epsilon} = \frac{\epsilon}{\dim_{\mathbb{R}} \mathbb{R}_{n(d+K_{\epsilon})}[X_1, \dots, X_n]} = \epsilon \binom{n(d+K_{\epsilon}+1)}{n(d+K_{\epsilon})}^{-1}$$

is allowed on each coefficient so that the corresponding approximated polynomial $\tilde{q}^{[\epsilon]} = \sum_{j=1}^m \tilde{u}_j^{[\epsilon]} p_j$ satisfies $|\tilde{q}^{[\epsilon]}(z)| \in [1 - 2\epsilon, 1 + 2\epsilon]$ for $z \in \overline{\mathbb{D}}^n$.

### 3. ABOUT THE POLYDISK MEMBERSHIP PROBLEM

Let $A_n$ be the (complexified) integral domain of structural stable $n$-dimensional systems :

$$A_n := \left\{ \frac{p}{q} : p, q \in \mathbb{C}[X_1, \dots, X_n], \zeta \in \overline{\mathbb{D}}^n \implies q(\zeta) \neq 0 \right\}.$$

Let $I \subset A_n$ be a finitely generated ideal of the integral domain $A_n$, namely $I = \sum_{j=1}^m p_j A_n$ for some $p_j$ in $\mathbb{C}[X_1, \dots, X_n]$.

Instead of assuming that $p_1, \dots, p_m$ do not share any common zero in the closed unit polydisk as in §2, one assumes in this section that $\bigcap_{j=1}^m p_j^{-1}(\{0\})$ hits $\overline{\mathbb{D}}^n$ and considers then as inputs, in addition to the $p_j$'s which generate $I$, a non-constant polynomial $p \in \mathbb{C}[X_1, \dots, X_n]$ which, considered as an element of $A_n$, belongs to $I$, which means that there exists $q \in \mathbb{C}[X_1, \dots, X_n]$ such that $q(\zeta) \neq 0$ for any $\zeta \in \overline{\mathbb{D}}^n$ and $u_1, \dots, u_m \in \mathbb{C}[X_1, \dots, X_n]$ such that $qp = \sum_{j=1}^m u_j p_j$ in $\mathbb{C}[X_1, \dots, X_n]$.
Let $\mathbb{I} = \sum_{j=1}^{m} p_j \mathbb{C}[X_1, \dots, X_n]$ and

$$ \mathrm{Ass} \left( \frac{\mathbb{C}[X_1, \dots, X_n]}{\mathbb{I}} \right) = \{ \mathcal{Q}_1, \dots, \mathcal{Q}_M \} \cup \mathcal{E}(\mathbb{C}^n \setminus \overline{\mathbb{D}}^n) $$

be the splitting of the list of associated primes of $\mathbb{C}[X_1, \dots, X_n]/\mathbb{I}$ between those (namely $\mathcal{Q}_1, \dots, \mathcal{Q}_M$) which zero set hits $\overline{\mathbb{D}}^n$ and the remaining ones, namely

$$ \mathcal{E}(\mathbb{C}^n \setminus \overline{\mathbb{D}}^n) = \emptyset \text{ or } \{\mathcal{Q}_1, \dots, \mathcal{Q}_N\}, $$

with $(\mathcal{Q}_v)^{-1}(\{0\}) \cap \overline{\mathbb{D}}^n = \emptyset$ for $v = 1, \dots, N$. Let

$$ (18) \qquad \mathbb{I} = (p_1, \dots, p_m) \mathbb{C}[X_1, \dots, X_n] = \left( \bigcap_{\mu=1}^{M} \mathcal{P}_{\mu} \right) \cap \left( \bigcap_{\nu=1}^{N} \mathcal{P}_{\nu} \right) $$

be then a primary decomposition of the ideal $\mathbb{I}$ in $\mathbb{C}[X_1, \dots, X_n]$, that is $\sqrt{\mathcal{P}_{\mu}} = \mathcal{Q}_v$ for $v = 1, \dots, M$, and, in case card$(\mathcal{E}(\mathbb{C}^n \setminus \overline{\mathbb{D}}^n)) = N \ge 1$, $\sqrt{\mathcal{P}_v} = \mathcal{Q}_v$ for $v = 1, \dots, N$. Despite the list of distinct associated primes $\mathcal{Q}_1, \dots, \mathcal{Q}_M, \mathcal{Q}_1, \dots, \mathcal{Q}_N$ is uniquely determined by the ideal $\mathbb{I}$, hence by the $p_j$'s, such is not the case for a primary decomposition of $\mathbb{I}$ as (18), due to the fact that there may be some embedded primes besides isolated ones among the total list of associated primes of $\mathbb{C}[X_1, \dots, X_n]/\mathbb{I}$.

In case card$(\mathcal{E}(\mathbb{C}^n \setminus \overline{\mathbb{D}}^n)) = N \ge 1$, let, for each $v$, $\{p_{v,\ell} : \ell = 1, \dots, L_v\}$ be a system of generators of $\mathcal{Q}_v$ in $\mathbb{C}[X_1, \dots, X_n]$. Since one has $\mathcal{Q}_v^{-1}(\{0\}) \cap \overline{\mathbb{D}}^n = \emptyset$ for each $v = 1, \dots, N$, one may invoke the method described in §2 and construct, for each $0 < \epsilon \ll 1$ and $v = 1, \dots, N$, an explicit polynomial $q_v^{[\epsilon]} \in \mathcal{Q}_v$ such that

$$ \forall z \in \overline{\mathbb{D}}^n, \quad 1 - \epsilon \le |q_v^{[\epsilon]}(z)| \le 1 + \epsilon. $$

The *polydisk membership problem* addresses the following question: given $p \in \mathbb{C}[X_1, \dots, X_n]$ which belongs to $\mathbb{I}$ when considered as an element of $A_n$, find explicitly $q \in \mathbb{C}[X_1, \dots, X_n]$ with $\tilde{\zeta} \mapsto |q(\tilde{\zeta})|$ uniformly arbitrarily close to 1 on $\overline{\mathbb{D}}^n$ and $u_1, \dots, u_m \in \mathbb{C}[X_1, \dots, X_n]$ such that $qp = \sum_{j=1}^m u_j p_j$ in $\mathbb{C}[X_1, \dots, X_n]$.

Let us start with two reformulations of the hypothesis $p \in \mathbb{I}$.

The first one involves the input $p \in \mathbb{C}[X_1, \dots, X_n]$ since it relies on the introduction of the carrier of the principal ideal $\mathbb{C}[X_1, \dots, X_n]$ of $\mathbb{C}[X_1, \dots, X_n]$ into the ideal $\mathbb{I}$, namely

$$ [p \mathbb{C}[X_1, \dots, X_n] : \mathbb{I}] := \{u \in \mathbb{C}[X_1, \dots, X_n] : up \in \mathbb{I}\}. $$

One can state then the following trivial equivalence.

**Proposition 3.** $p \in \mathbb{I} \iff 1 \in [p \mathbb{C}[X_1, \dots, X_n] : \mathbb{I}] A_n$.

*Proof of Proposition 3.* If $p \in \mathbb{I}$, there is $q \in \mathbb{C}[X_1, \dots, X_n]$ such that $q(\tilde{\zeta}) \ne 0$ for any $\tilde{\zeta} \in \overline{\mathbb{D}}^n$ and $u_1, \dots, u_m \in \mathbb{C}[X_1, \dots, X_n]$ such that $1 \times p = (1/q) \sum_{j=1}^m u_j p_j \in \mathbb{I}$, which means that $1 \in [p \mathbb{C}[X_1, \dots, X_n] : \mathbb{I}] A_n$. Conversely, if $1 \in [p \mathbb{C}[X_1, \dots, X_n] : \mathbb{I}] A_n$, one has

$$ p \in p [p \mathbb{C}[X_1, \dots, X_n] : \mathbb{I}] A_n \subset \mathbb{I} A_n = \mathbb{I}. $$

□
The alternative reformulation of $p \in \mathcal{I}$ relies on the choice of a primary decomposition such as (18) for $\mathcal{I}$, hence is independent on $p \in \mathcal{I}$.

**Proposition 4.** $p \in \mathcal{I} \iff p \in \bigcap_{\mu=1}^{M} \mathcal{P}_{\mu}$.

*Proof of Proposition 4.* One has

$$p \in \mathcal{I} \implies \forall z \in \overline{\mathbb{D}}^n, p \mathcal{O}_z \subset \sum_{j=1}^{m} p_j \mathcal{O}_z \\ \text{(that is } \forall z \in \overline{\mathbb{D}}^n, \forall \mu = 1, \dots, M, p \mathcal{O}_z \subset \mathcal{P}_{\mu} \mathcal{O}_z),$$

where $\mathcal{O}$ denotes the structural sheaf of $\mathbb{C}^n$. The membership of $p$ (considered in $\mathbb{C}[X_1, \dots, X_n]$) to each primary ideal $\mathcal{P}_\mu$ is characterized by the vanishing on the irreducible algebraic variety $\mathcal{Q}_\mu^{-1}(\{0\})$ of the action on $p$ of a finite list of nœtherian differential operators with polynomial coefficients

$$\{A_{\mu, \iota}(\xi, \partial/\partial\xi_1, \dots, \partial/\partial\xi_n) : \iota \in I_{\mu}\}.$$

On the other hand, for any $\mu = 1, \dots, M$, the set $(\mathcal{Q}_\mu^{-1}(\{0\}))_{\text{reg}}$ of smooth points of $(\mathcal{Q}_\mu)^{-1}(\{0\})$ is open, dense and connected in $(\mathcal{Q}_\mu)^{-1}(\{0\})$ since any Zariski open subset of a projective subvariety $\mathcal{X}$ of $\mathbb{P}^n(\mathbb{C})$ is connected for the usual topology. It follows then from the analytic continuation principle applied on each complex manifold $(\mathcal{Q}_\mu^{-1}(\{0\}))_{\text{reg}}$ that if $p \mathcal{O}_z \subset \mathcal{P}_\mu \mathcal{O}_z$ for any $z \in \overline{\mathbb{D}}^n$ and any $\mu = 1, \dots, M$, then $p \in \bigcap_{\mu=1}^{M} \mathcal{P}_\mu$. The direct implication in Proposition 4 is proved. In case $\mathcal{E}(\mathbb{C}^n \setminus \overline{\mathbb{D}}^n) = \emptyset$, we are done. Otherwise, one considers for each $v = 1, \dots, N$, the smallest integer $e_v \in \mathbb{N}$ such that $(\mathcal{Q}_v)^{e_v} \subset \mathcal{P}_v$. Then one has for any $0 \le \epsilon \ll 1$ that

$$(19) \qquad p \in \bigcap_{\mu=1}^{M} \mathcal{P}_{\mu} \implies \left( \prod_{v=1}^{N} (q_v^{[\epsilon]}(z))^{e_v} \right) p \in \mathcal{I},$$

from which $p \in \mathcal{I}$ follows. Observe that, provided one takes $0 < \epsilon \ll 1$ small enough, one can even ensure that

$$\forall \xi \in \overline{\mathbb{D}}^n, 1 - \eta \le \left| \prod_{v=1}^{N} (q_v^{[\epsilon]}(\xi))^{e_v} \right| \le 1 + \eta,$$

where $0 < \eta \ll 1$ is arbitrary. $\square$

<figure><img src="image_2.png" /></figure>

**Example.** When $m \le n$ and $p_1, \dots, p_m$ are such that

$$\mathrm{codim}_{\mathbb{C}^n} \{ \xi \in \mathbb{C}^n : \forall j = 0, \dots, m, p_j(\xi) = 0 \} = m,$$

all associate primes of $\mathbb{C}[X_1, \dots, X_n]/\mathcal{I}$ are isolated, that is are minimal elements in $\mathrm{Ass}(\mathbb{C}[X_1, \dots, X_n]/\mathcal{I})$ with respect to the inclusion ordering. A typical example is that of $n = 3$, $m = 2$, where $p^{-1}(\{0\})$ is an algebraic curve. One has in this case the irreducible decompositions in $\mathbb{C}[X_1, X_2, X_3]$:

$$p_1 = \prod_{\kappa=1}^{s} (p_{1,\kappa})^{e_{1,\kappa}}, \quad p_2 = \prod_{\kappa'=1}^{s'} (p_{2,\kappa'})^{e_{2,\kappa'}}$$

where the set of common zeroes of each pair $(p_{1,\kappa}, p_{2,\kappa'})$ is an irreducible algebraic curve in $\mathbb{C}^3$ and multiplicities $e_{1,\kappa}, e_{2,\kappa'}$ are strictly positive integers. For each $0 < \epsilon \ll 1$ and each pair $(\kappa, \kappa')$ such that

$$\{\xi \in \mathbb{C}^3 : p_{1,\kappa}(\xi) = p_{2,\kappa'}(\xi) = 0\} \cap \overline{\mathbb{D}}^3 = \emptyset,$$
let us say $(\kappa, \kappa') \in \mathcal{K}$, one can construct as in §2 a polynomial

$$q_{\kappa, \kappa'}^{[\epsilon]} \in p_{1,\kappa} \mathbb{C}[X_1, X_2, X_3] + p_{2,\kappa'} \mathbb{C}[X_1, X_2, X_3]$$

such that $\xi \mapsto |q_{\kappa,\kappa'}(\xi)|$ is arbitrarily close to 1 in $\overline{\mathbb{D}}^3$. It follows then from (Jelonek, 2025, Theorem 1.3) that

$$(20) \qquad \left( \prod_{(\kappa, \kappa') \in \mathcal{K}} (q_{\kappa, \kappa'}^{[\epsilon]})^{e_{1,\kappa} e_{2,\kappa'} \deg p_{1,\kappa} \deg p_{2,\kappa'}} \right) p \in \mathbb{I}$$

with, given $0 < \eta \ll 1$ arbitrarily small,

$$1 - \eta \le \left| \prod_{(\kappa, \kappa') \in \mathcal{K}} (q_{\kappa, \kappa'}^{[\epsilon]})^{e_{1,\kappa} e_{2,\kappa'} \deg p_{1,\kappa} \deg p_{2,\kappa'}} \right| \le 1 + \eta$$

in $\overline{\mathbb{D}}^3$ provided $\epsilon > 0$ is chosen close enough to 0 (depending on $\epsilon$). Observe that, despite the fact that the polynomial membership decision problem over $\mathbb{C}$ is known to be EXPSPACE-complete in view of the example explicit in Mayr-Meyer (1982), the membership to the polynomial ideal $\mathbb{I}$ in (20) can be realized explicitly in the particular case considered in this example through Jacobi-Lagrange-Kronecker parametric identities, see (Vidras-Yger, 2023, Theorem 7.34). It is also important to point out that the weaker assumption $|p|/\|\mathbf{p}\| \in L^\infty(\overline{\mathbb{D}}^3)$ (instead of $p \mathcal{O}_z \in p_1 \mathcal{O}_z + p_2 \mathcal{O}_z$ for any $z \in \overline{\mathbb{D}}^3$) implies that the membership

$$(21) \qquad \left( \prod_{(\kappa, \kappa') \in \mathcal{K}} (q_{\kappa, \kappa'}^{[\epsilon]})^{e_{1,\kappa} e_{2,\kappa'} \deg p_{1,\kappa} \deg p_{2,\kappa'}} \right) p^2 \in \mathbb{I}$$

holds as a weaker substitute to (20) and moreover can be realized explicitly through an approach similar to that which was used in §2 in order to express in $\overline{\mathbb{D}}^3$ the function 1 in the polynomial ideal generated by $p_1, p_2$. Such fact still holds in the general case where $m \le n$ and the $p_j$'s satisfy $\mathrm{codim}_{\mathbb{C}^n}\{\xi \in \mathbb{C}^n : \forall j = 0, \dots, m, p_j(\xi) = 0\} = m$, for what concerns this time the explicitation of the membership

$$(\prod_{\nu=1}^{N} (q_{\nu}^{[\epsilon]}(z))^{e_{\nu}})^{m} \in \sum_{j=1}^{m} p_j \mathbb{C}[X_1, \dots, X_n]$$

provided that the analytic condition $|p|/\|\mathbf{p}\| \in L^\infty(\overline{\mathbb{D}}^n)$ holds (instead of $p \mathcal{O}_z \in \sum_{j=1}^m p_j \mathcal{O}_z$ for any $z \in \overline{\mathbb{D}}^n$), see e.g. (Vidras-Yger, 2023, Theorem 7.47). Here,

$$N, q_{\nu}^{[\epsilon]}, e_{\nu} \quad (\nu = 1, \dots, N)$$

are still defined as before the implication (19).

While the polydisk nullstellensatz problem, which solution is explicit in §2, appears constantly in control theory since explicitation of controllers lead naturally to it, see Cluzeau-Moroz-Quadrat (2024), the polydisk membership problem is related to inverse problems such as for example the realization of deconvolvers in multi-filtering. It is therefore important to observe that the solution of the polydisk nullstellensatz on which it relies is based on Wiener filtering as a starting point, see the final remark in §2.

## REFERENCES

K. Adachi, M. Andersson, H.R. Cho. $L^p$ and $H^p$ extensions of holomorphic functions from subvarieties of analytic polyhedra. *Pacific J. of Math.*, 189, 2: 201–210, 1999.
M. Andersson. $L_p$ estimates for the $\bar{\partial}$ equation in analytic polyhedra in Stein manifolds. In *Proc. of the Mittag-Leffler Inst. 87/88*, pages 34–47, Math. Notes 38, Princeton Univ. Press, 1992.

M. Andersson, B. Berndtsson. Henkin-Ramirez formulas with weight factors. *Ann. Fourier*, 32, 3: 91–110, 1982.

M. Andersson, M. Passare. A shortcut to weighted representation formulas for holomorphic functions. *Ark. Math.*, 26: 1–12, 1988.

C.A. Berenstein, R. Gay. *Complex Variables, an introduction*, GTM 125. Springer-Verlag, 1991.

C.A. Berenstein, R. Gay, A. Vidras, A. Yger. *Residue currents and Bézout identities*, Prog. in Math. 114 Birkhäuser, 1993.

Y. Bouzidi, A. Quadrat, F. Rouillier. Certified non-conservative tests for the structural stability of discrete multidimensional systems. *Mult. Syst. Signal Process.*, 30, 3: 1205-1235, 2019.

D. Bridges, R. Mines, F. Richman, P. Schuster. The polydisk nullstellensatz. *Proceedings of the American Math. Society*, 132, 7: 2133–2140, 2003.

T. Cluzeau, G. Moroz, A. Quadrat. Towards the Computation of Stabilizing Controllers of Multi-dimensional Systems. IFAC-PapersOnLine 58, 17: 433–438, 2024.

Z. Jelonek. On the effective Nullstellensatz. *Invent. Math.*, 162, 1: 1–17, 2005.

E. W. Mayr, A. Meyer. The complexity of the word problem for commutative semi-groups and polynomial ideals. *Adv. in Math.*, 46, 3: 305-329, 1982.

A. Vidras, A. Yger. *Multidimensional residues and applications*, Math. surveys and monographs 275. AMS, Providence, Rhode Island, 2023.

UNIVERSITÉ DE REIMS CHAMPAGNE ARDENNE, CRESTIC, FRANCE

Email address: Mamadou.Mboup@univ-reims.fr

IMB, UNIVERSITÉ DE BORDEAUX, 33405, TALENCE, FRANCE

Email address: yger@math.u-bordeaux.fr