Corrections and additions to "A note on the hit problem for the polynomial algebra in the case of odd primes and its application" [Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 118 (2024), article 22]
arXiv:2510.17908 · doi:10.1007/s13398-023-01517-4
Abstract
In this work, we correct results in the published paper [Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 118 (2024), article 22]. We determine the non-hit kernel of the first reduced power in polynomial degree , identify the published Frobenius construction as zero on indecomposables, and replace the multiplier assertion by a theorem with explicit degree hypotheses. We also classify the monomial multipliers of Frobenius that preserve hits in one variable. For the third algebraic transfer, we correct the target dimensions and establish surjectivity throughout the four principal degree families. Explicit invariant calculations and Minami's degree reduction give isomorphisms in seven infinite families at and , including a family with an initial change from dimension one to dimension two. For every odd prime, compatibility with the Frobenius operation on gives an exact sequence comparing successive transfer kernels and forces a nonzero invariant extraction kernel at every second-family dimension jump. We compute the first cohomology of a nonzero extraction kernel and determine the exact obstruction to injectivity from rank-two products in the first and fourth families. Coefficient extraction and changes of coordinates give an additional infinite isomorphism family and a spanning result for annihilated polynomial homology. At each extraction isomorphism, the quotient by all rank-two products is preserved as a general linear group module. We distinguish these results from the unresolved invariant dimensions at arbitrary normalized index differences, and give the precise limitations of composition-factor counts and Dickson multiplication. An appendix supplies explicit representatives for all the nonzero rank-two invariant lines in Crossley's dimension classification.
arXiv comment62 pages. This new version presents corrections and additions to our 2024 RACSAM article, including corrected statements and detailed proofs. It also provides a comprehensive response to Nguyen Sum's comments (arXiv:2502.11119)