Source, priority and significance assessment

Henry Zweiman. September 15, 2026.

Exact target

Saini--Singh, arXiv:2604.27592v1, Question 6.1, asks whether n>k(nullity(B)-1) suffices for surjectivity of X^k+B Y^k over algebraically closed characteristic-zero fields. The authors prove necessity, sufficiency for n<=4, and the all-dimension nullity-at-most-one case. The current arXiv history, inspected September 15, lists only v1, April 30, 2026, and describes the work as a preliminary version.

The new family has nullity exactly two, n=k+1 and k>=5. It satisfies the proposed sufficient inequality while missing an explicitly specified rank-three target. Thus the manuscript addresses the full yes/no sufficiency question by a negative answer, rather than proving a restricted positive subcase. The counterexamples have a diagonal idempotent coefficient and work over every field.

Comparison with related work

The primary Saini--Singh introduction, Theorems 1.2/1.3, reductions, Proposition 4.2, rank criterion, and conclusion were reviewed against the target. The manuscript preserves their established necessity and positive cases. Its obstruction constrains both summands, beyond the first-summand nilpotent-block obstruction used for necessity.

Gangwal--Mandal--Verma, arXiv:2607.27226v1, Question 1.1, asks for surjectivity and image linearity over division algebras. The inspected introduction and Theorems 1.3/1.4 treat matrices of size two over the indicated real, complex and quaternion settings. They do not supply the present family of size at least six. No complete review of every theorem in that paper is claimed.

P33 in this repository concerns a uniform polynomial-completion theorem and coefficient column spaces forming a direct sum. With one coefficient equal to identity and another nonzero, the column spaces cannot form a direct sum. P39 therefore answers a distinct previously unresolved question. It is not a renamed corollary of P33, and its proof does not use P33.

The cyclic nilpotent argument and elementary kernel/image facts are standard linear algebra. Novelty is claimed for their application producing the explicit counterexample family, not for Jordan theory or the density of invertible matrices. The nonadditivity corollary is part of this same paper.

Bounded priority search

Searches on September 15 for the exact arXiv identifier with counterexample/surjectivity, the authors with matrix powers and nullity, and the proposed rank obstruction did not locate a matching primary result. Some results were irrelevant or secondary; they were not used as proof of novelty. The primary arXiv history has no later version. A guessed author-page route returned an error and supplied no evidence. This is a current, bounded source check, not a proof that no earlier result exists anywhere.

There is no dependency here on the inaccessible corrected Kishnani--Kulshrestha article that affects the separate group-theory investigations. No author was contacted and no paywalled text was bypassed.

Significance and limits

The principal contribution is an explicit infinite counterexample family to a recent stated criterion, with the smallest possible positive nullity beyond the known nullity-one theorem. It shows that coefficient nullity together with the proposed dimensional inequality is insufficient, even for a semisimple coefficient. The two-summand obstruction may help formulate a replacement criterion. This is a focused short paper, not a solution of the entire matrix Waring or multilinear image problem.

The manuscript does not claim a dimension-minimal counterexample, a classification at fixed small k, all five-dimensional cases, or a description of the full image. No citation-count forecast is justified. Internal proof checks support publishing a proposed result; human expert verification, exhaustive priority certification, journal acceptance, and long-term significance remain unestablished.

Admission decision: prepare and publish this as one internally audited proposed paper, with the source/proof limitations visible. Do not treat its publication or the collection's manuscript count as certification that the 45-paper research goal is complete.