Conjecture on tangent-space inclusion for singular real algebraic hypersurfaces

About 1 year old · traced to 2025

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.

References

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).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An unverified posted example claims to disprove the conjecture, while the published theorem covers only a restricted class of polynomial systems.

Basu and Mohammad-Nezhad (2025) conjectured that the inclusion $T_{\bar{x}}Z\subseteq\lim_{\xi}(T_{x_\xi}V_\xi)$ remains valid without their general-position hypothesis.

Known results

Basu and Mohammad-Nezhad (2025) proved the tangent-space inclusion for polynomial families in general position, as part of their critical-point convergence theorem.

Posted attempt

A proposed example takes $P(X,Y)=X^2(X^2+Y^2)$ and claims that, at the origin, the canonical stratum has tangent space $\mathbb{R}(0,1)$ while the limiting perturbed tangent space satisfies $3u+v=0$ , disproving the inclusion even for a smooth real zero locus. This is a complete counterexample claim, but it has not been independently verified.

Current status (as of August 2026): The inclusion is established under the general-position hypothesis; a posted counterexample claims to settle the unrestricted conjecture, but that claim remains unverified.

Sources

arxiv.org

CounterexampleThis solution needs a summarySee full solutionHide full solution

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.