Recover the nonnormal image algebra from polarized normalization data

Determine the actual possibly nonnormal image algebra of a sparse divisor or profile image from its normalization and its natural polarizations, using whatever additional information is required beyond the normalized geometry.

Background

The paper develops explicit normalizations of relation branches and, for complete progressions, explicit product descriptions of image normalizations together with two natural polarization classes. These data describe the normalized geometry and provide normality criteria, but they do not encode the full coordinate algebra of an image when that image is nonnormal.

The unresolved issue is therefore to recover the original, potentially nonnormal image algebra from its normalization and polarizations. The paper explicitly states that such recovery requires information beyond the normalization data and is not pursued within the present work.

References

Once the normal profile image and its polarizations are known, a finer question is to recover the actual, possibly nonnormal, image algebra from its normalization. Such a recovery requires information beyond normalization and its natural polarizations and is not pursued here.

— Reconstruction of Binary Linear Systems and Profile Geometry of Sparse Krylov Strata  (2609.10015 - Li, 9 Sep 2026) in Section 1, Introduction and main results; subsection “The normalization results mark a natural boundary of the present paper”