Conjecture on tangent-space inclusion for singular real algebraic hypersurfaces

From papers

Let $\mathbb{R}\langle \xi\rangle$ be the real closed field of algebraic Puiseux series, let $\mathcal{P}=\{P_1,\ldots,P_s\}\subset \mathbb{R}[X_1,\ldots,X_n]$ , and set

V=\{x\in\mathbb{R}^n:\prod_{i=1}^sP_i(x)=0\}.

For a bounded point $x_\xi\in V_\xi=\{x\in\mathbb{R}\langle\xi\rangle^n:\prod_{i=1}^sP_i(x)=\xi\}$ with $\bar{x}=\lim_\xi x_\xi$ , let $T_{x_\xi}V_\xi$ denote the tangent space at $x_\xi$ , and let $\lim_\xi(T_{x_\xi}V_\xi)$ denote its limit as $\xi$ tends to zero. Tangent-space inclusion conjecture. The inclusion

holds. The surrounding discussion notes that this extends the tangent-space result beyond the general-position case, including situations in which $\prod_iP_i$ has finitely many singular zeros. The conjectured extension concerns more general singularities and is presented without a resolution.

Progress summary

Open

The conjecture remains open beyond the special configurations already covered by the existing theorem, with no verified proof or counterexample found.

The conjecture asks whether the tangent space of the canonical Whitney stratum through the limiting point is contained in the limit of tangent spaces to the perturbed hypersurfaces. The broader statement extends a result previously proved only under a general-position assumption.

Known results

Basu and Mohammad-Nezhad (2025) proved the corresponding critical-point and tangent-space limit theorem when the polynomial family is in general position, identifying the limiting critical point relative to the canonical Whitney stratification.

June 2025 conjecture statement

Basu and Mohammad-Nezhad explicitly conjectured that their theorem remains valid without the general-position hypothesis. The scan found no published proof, counterexample, withdrawal, referee report, or other verified resolution of this extension.

Current status (as of August 2026): The inclusion is established under the general-position hypothesis, while the stated extension to arbitrary singular configurations remains open.

Sources

Sources & referencesView supporting material

Primary source

Saugata Basu and Ali Mohammad-Nezhad, “On the convergence of critical points on real algebraic sets and applications to optimization”, arXiv:2506.20565 (2025).

1 week ago

Take $s=1$ , $n=2$ , and

P(X,Y)=X^2(X^2+Y^2).

Its real zero locus is the smooth line

V=\{(0,y):y\in\mathbb R\},

whose real vanishing ideal is $\mathcal I(V)=(X)$ . Therefore $\operatorname{Ireg}(V)=\varnothing$ , so the canonical stratification of Definition 4.6 has the single stratum $Z=V$ . At $\bar x=(0,0)$ ,

T_{\bar x}Z=\mathbb R(0,1).

For a positive infinitesimal $\xi$ , put

t=(\xi/2)^{1/4},\qquad x_\xi=(t,t).

Then $x_\xi$ is bounded, $\lim_\xi x_\xi=\bar x$ , and

P(x_\xi)=2t^4=\xi,

hence $x_\xi\in V_\xi$ . Moreover,

\nabla P(x_\xi)=(6t^3,2t^3)\ne0.

Consequently the bounded tangent space in the conjecture is

T_{x_\xi}V_\xi =\{(u,v):3u+v=0,\ u^2+v^2\le1\}.

This set is independent of $\xi$ and therefore equals its own limit. But

(0,1)\in T_{\bar x}Z, \qquad (0,1)\notin\lim_\xi(T_{x_\xi}V_\xi).

Thus

T_{\bar x}Z\not\subseteq\lim_\xi(T_{x_\xi}V_\xi),

disproving the conjecture even when the real zero locus is smooth and its canonical stratification is genuinely Whitney.

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Shivam Patel · 1 day ago