Gradient-Tentacle Certificates for Polynomial Nonnegativity

Definitions and mathematical statement

Let $A={\mathbb{R}}[X_1,\ldots,X_n]$, let $f\in A$, and write $f_i=\partial f/\partial X_i$. Put $f^*=\inf_{x\in{\mathbb{R}}^n}f(x)$, without an assumption that the infimum is attained. The cleaned statement defines \[ J=J_{\mathrm{skew}}=(X_jf_i-X_if_j:1\leq i<j\leq n). \] Let $V_{\mathbb{R}}(J)$ be its common real zero set, let $\Sigma A^2$ be the finite sums of polynomial squares, and let $\sqrt J=\{h\in A:h^r\in J\text{ for some }r\geq1\}$ be the ordinary algebraic radical, not silently the real radical.

The cleaned source asks whether the following four conditions are equivalent for every indicated polynomial, with no attained-minimum hypothesis: \[ \text{(i)}\quad f(x)\geq0\quad(x\in{\mathbb{R}}^n), \] \[ \text{(ii)}\quad f(x)\geq0\quad(x\in V_{\mathbb{R}}(J)), \] \[ \text{(iii)}\quad f\in\Sigma A^2+\sqrt J, \] \[ \text{(iv)}\quad\forall\varepsilon>0,\quad f+\varepsilon\in\Sigma A^2+J. \] In (iv), the sum-of-squares representation and ideal multipliers may depend on $\varepsilon$; no common degree bound is requested. The exact zero-level certificate in (iii) and the positive-perturbation certificates in (iv) are separate clauses. The source does not explicitly require $f^*$ finite, so no bounded-below hypothesis is inserted beyond the proposed conditions themselves.

There is a material generator discrepancy: the original wording describes the ideal generated by $X_jf_i$ and $X_if_j$ separately, whereas the cleaned version uses their differences. The literal separate-product ideal would be \[ J_{\mathrm{sep}}=(X_jf_i,\ X_if_j:1\leq i<j\leq n), \] which need not equal $J_{\mathrm{skew}}$. For example, for $f=X_1^2+X_2^2$, the first difference vanishes identically, while the separate products generate $(X_1X_2)$. This diagnoses inequivalent readings, not a counterexample to the proposed four-way statement. The supplied source-verification claim does not explain the discrepancy, so the exact original ideal must be reconciled rather than silently chosen.

The supplied literature triage checked 2026-08-21 attributes the replacement conjecture to Schweighofer and reports generic and special-case progress but no unrestricted resolution in its assessment. No exact partial-result hypotheses or proof of any proposed equivalence is supplied.

Short English statement

The curated question proposes four equivalent nonnegativity and sum-of-squares conditions after replacing the gradient ideal by the displayed tangency ideal, without assuming a minimum is attained. The original text instead names separate-product generators, a substantive discrepancy requiring source reconciliation.

Sources

• Local UnsolvedMath registry, permanent ID 30000717, catalogue entry, Gradient-Tentacle Certificates for Polynomial Nonnegativity; the supplied assessment attributes the conjecture to Schweighofer.
• Reelle Algebraische Geometrie (2007), Oberwolfach Reports, pp. 753--820, DOI https://doi.org/10.4171/owr/2007/14.
• Supplied reference "gradient tentacles paper": https://arxiv.org/abs/math/0511617.
• Supplied verification claims comparison with the official report; supplied literature triage checked 2026-08-21 does not reconcile the two generator descriptions or state exact special-case hypotheses.