Birch’s theorem and the variable-bound problem for the smooth Hasse principle

The problem concerns lowering the number of variables needed for the smooth Hasse principle for degree-$d$ forms, particularly when $d\ge 5$. The retrieved sources record substantial earlier improvements but do not identify a final optimal threshold.

Known results

• An earlier theorem establishes the smooth Hasse principle when $n-\sigma\geq\left(d-\frac12\sqrt d\right)2^d$.
• For $d\in\{3,4,5,6,7,8,9\}$, a stronger condition is $n-\sigma>\frac34d2^d-2d$.
• These bounds include nonsingular cubic forms in at least $13$ variables and save $18$ variables over Birch’s bound for quintic forms.

September 2026 claimed improvement

A September 2026 preprint, Improved bounds in Birch's theorem for forms in many variables, reports an improvement to the best known variable bound for degrees $d\ge 5$. The numerical bound is not given in the retrieved abstract, and the claim is unrefereed and therefore unverified.

Current status (as of September 2026): Earlier explicit bounds are established, while a newer improvement for $d\ge 5$ is claimed but unverified; the optimal variable-bound problem remains open.