Mathematical scope and priority assessment

Date: September 9, 2026.

Target and contribution

Gangwal, Mandal and Verma, arXiv:2607.27226v1, Question 1.1, ask for necessary and sufficient conditions for surjectivity and image linearity of AX^k+BY^l over infinite division algebras. Their results concern two by two real and quaternion matrices. Panja, Saini and Singh had treated the algebraically closed two by two case in Mathematika 71 (2025), e70031.

The present manuscript gives a complete all-dimension classification over an algebraically closed characteristic-zero field for coefficient families whose column spaces form a direct sum. Its central theorem is the sharp uniform completion criterion n > E(f)(r-1), for arbitrary nonconstant scalar polynomials. The new proposed contribution is sufficiency and the resulting classification, not the familiar Jordan-block obstruction.

This is a substantial restricted regime of a published broad question. It is not a full resolution of that broad question. The exact prior open status of the standalone polynomial completion formulation is not independently certified. Targeted searches for one-sided polynomial matrix equations, prescribed-row powers, and matrix completion found no earlier matching threshold theorem. A search failure is not an exhaustive priority certificate.

Distinguishing neighboring results

Saini--Singh arXiv:2604.27592v1 establish a necessary dimension bound for X^k+B Y^k and ask in Question 6.1 whether it is sufficient. Their Proposition 4.2 contains the nilpotent obstruction used here. The current manuscript proves the relaxed X^k+B Y criterion, but it does not replace Y by a k-th power. It therefore does not solve their Question 6.1. Both 2026 arXiv records list only v1 in the current audit.

The older surjectivity conjecture in Panja--Saini--Singh is already contradicted by the necessary condition in Saini--Singh. No new priority claim for disproving that conjecture is made. Nor is there a claim to solve the classical or generalized multilinear L'vov--Kaplansky conjecture.

The partial-operator chain decomposition is a classical special case of Kronecker canonical form, and is explicitly credited to that theory. Verdier's primary preprint, Section 8, was read for the block decomposition and attribution. A direct filtration proof is supplied, so this manuscript does not depend on unexamined proof steps in the recent polynomial-map papers.

Wang--Feng--Huang, Electronic Research Archive 29 (2021), 2249--2267, DOI 10.3934/era.2020114, studies AX^2=B with the semi-tensor product and variable compatible dimensions. Its abstract, Theorem 3.8, and the reduction/algorithm in Section 3.2.2 were checked. That source provides related matrix-equation methods, not the uniform rank threshold claimed here. This was a focused comparison, not a full audit of every semi-tensor result.

Value and limits

The completion theorem is reusable independently of the coefficient application. It shows that the relevant parameter is minimum ramification in an exceptional fiber, rather than the degree of the polynomial. The direct-sum criterion applies to arbitrary coefficient matrices with those images, not merely complementary projections. This may support further work on polynomial interpolation of operators and coefficient maps with intersecting images.

Independent expert correctness review, broader historical-priority investigation and significance assessment remain outstanding. No citation forecast, journal acceptance, or complete solution of the overlapping-column-space problem is claimed. This counts as one manuscript; its corollaries are not separate papers.