Polynomial-size four-square certificates for real emptiness

Establish whether every real-empty finite algebra obtained from the augmented critical-point system \(G_i=0\) and \(Y^8+J(X)-T=0\) admits a certificate of the form \(c^2+U_1^2+U_2^2+U_3^2+U_4^2\in(G_1,\ldots,G_m,Y^8+J-T)\), with nonzero integer \(c\) and total integer-polynomial-circuit size bounded by a fixed polynomial in the original ETR input length.

Background

The paper reduces real feasibility to the existence of a real point in a finite rational algebra obtained by adjoining the equation Y8+J(X)−T=0Y^8+J(X)-T=0 to the critical equations. A finite-dimensional rational algebra has no real points exactly when −1-1 is a sum of four squares in that algebra, which yields a certificate of real emptiness after clearing denominators.

The authors show that a supplied certificate can be verified in coRPPP and would imply a stronger conditional upper bound for ETR. What remains unresolved is whether such certificates can always be represented by polynomial-size integer arithmetic circuits; dense algebraic existence results do not establish this succinctness.

References

The circuit-size hypothesis is open here. Four elements can still require exponentially many coordinates in the standard basis.

— $\exists \mathbb{R} \subseteq \textsf{CH}$  (2610.10514 - Meiburg, 7 Oct 2026) in Appendix, Section 7, subsection “Real emptiness and four-square certificates,” Proposition 7.conditional

The hypothesis in Proposition~\ref{push9:conditional} remains open. Theorem~\ref{push9:dyadic} removes the unknown scalar denominator and supplies uniform dense bounds, but does not prove this hypothesis.

— $\exists \mathbb{R} \subseteq \textsf{CH}$  (2610.10514 - Meiburg, 7 Oct 2026) in Appendix, Section 9, subsection “Verifying a compressed family of squares,” Proposition 9.conditional

The description-size hypothesis is still unproved. The effective rational finite-set Positivstellensatz of Baldi, Krick and Mourrain supplies dense degree and height bounds, but its Gram matrix construction, rounding, and factorization have dimension polynomial in the full quotient dimension.

— $\exists \mathbb{R} \subseteq \textsf{CH}$  (2610.10514 - Meiburg, 7 Oct 2026) in Appendix, Section 8, subsection “Allowing weighted and coefficient-defined certificates”