Algebraic Complexity Bounds for Real Varieties

Definitions and mathematical statement

Let $Y$ be a real algebraic variety of dimension $d\geq2$, and let $\beta(Y)$ denote the source's invariant. The main conjectural inequality is \[ \forall Y\text{ in the intended real-algebraic class},\qquad \beta(Y)\leq\dim Y-1. \] The record also asks for the exact sphere values \[ \beta(S^n)=n-1\qquad\text{for every integer }n\geq2, \] where the unit sphere carries its standard real algebraic model \[ S^n=\left\{(x_0,\ldots,x_n)\in{\mathbb{R}}^{n+1}:\sum_{i=0}^{n}x_i^2=1\right\}. \] The general inequality would give only the upper bound on spheres; equality additionally requires the matching lower bound, which is not supplied in the extract.

The essential invariant $\beta$ is not defined in any field of this record. Thus these formulas preserve the exact target and quantifiers but cannot be promoted to a fully specified assertion about a known invariant. In particular $\beta$ is not guessed to be a Betti number, a polynomial degree or an unspecified generic complexity measure. The class of varieties, dimension convention and definition of $\beta$ must be recovered from the supplied report and partial-results reference.

Short English statement

Ask whether the source's invariant $\beta(Y)$ is at most dimension minus one for every real variety of dimension at least two, with equality for every unit sphere of dimension at least two. The invariant itself is missing, so the target remains conditional on its recovery.

Sources

• Supplied report: Reelle Algebraische Geometrie (2007), Oberwolfach Reports, DOI 10.4171/owr/2007/14, pp. 753--820; proposed year 2007. https://doi.org/10.4171/owr/2007/14.
• Supplied reference labelled "Bochnak--Kucharz partial results": https://doi.org/10.1007/s00208-006-0061-3.
• Local UnsolvedMath registry, permanent ID 30000712, catalogue entry; created 25 August 2026, updated 16 September 2026. The corrected_verified note reports comparison with the official report. Its supplied 21 August 2026 literature triage labels the inequality and all-dimensional sphere equality open, reporting partial bounds rather than a general proof. This dated assessment does not define $\beta$ and is not independently updated here.