Let K be a field, let V=V_1×…× V_d, and let P:V→ K be a multilinear polynomial of degree d. Write r_K(P) for the Schmidt rank of P, and let bar K be the algebraic closure of K. For any d≥ 2, rank-comparison conjecture. there exists kappa_d>0 such that r_K(P)≤kappa_d r_bar K(P). The claim is known for d=2, with kappa_2=1/2, and for d=3 because Schmidt rank equals slice rank and the required comparison follows from the cited results. The general statement remains open.

Conjecture on rank comparison over a field and its algebraic closure

The conjecture asks whether Schmidt rank over any field can be bounded by a degree-dependent multiple of Schmidt rank over its algebraic closure. The cases $d=2$ and $d=3$ are known, but no result settles all degrees and fields.

Known results

$d=2$ : settled immediately.
$d=3$ : Schmidt rank equals slice rank; the source records $r_K(P)\leq \frac{3}{2}r_{\bar K}(P)$ (Dersken).
Quartics: over perfect fields of characteristic different from $2$ , a bound $r_K(P)\leq c(r_{\bar K}(P))$ is proved, but not a degree-only linear bound.
For number, finite, and function fields, polynomial bounds in $r_{\bar K}(P)$ are available for homogeneous forms.

January 31, 2024 field-restricted bounds

A revised paper gives polynomial field-restricted bounds and states that, before this work, no such bound was known in degrees $d>4$ . This is substantial progress but does not establish a constant $\kappa_d$ uniform over all fields.

Current status (as of October 2026): The conjecture is settled for $d=2$ and $d=3$ , has weaker or field-restricted higher-degree results, and remains open in general.

Conjecture on rank comparison over a field and its algebraic closure

Primary source

Progress summary

Solutions 0