Structure of the head of generalized eigenvectors

Determine how the initial entries of the generalized eigenvectors associated with the diagonal restriction of the multivariate Eulerian spectrahedral relaxation grow and use that structure to improve the approximating vector sequence.

Background

The proposed sequence of linearizing vectors was inferred from numerical generalized eigenvectors for small instances of the relaxation. The tail and transition region of these eigenvectors appeared comparatively structured, while the behavior of the first entries—the head—was not directly controlled.

The unresolved issue concerns the precise growth pattern in this head region, which is important for constructing more accurate generalized-eigenvector guesses and potentially sharper asymptotic root bounds.

References

Thus, this analysis only lets open the question of how the growth in the head happens.

— Guessing sequences of eigenvectors for LMPs defining spectrahedral relaxations of Eulerian rigidly convex sets  (2507.18434 - Nevado, 24 Jul 2025) in Experiment “Guessing the good sequence of vectors,” Section 8, “Guessing eigenvectors with care for effective multivariability”