Exact certificates for three open cases of the Heo-Kiem range-criterion problem: (m,n,k,l) = (5,5,3,5), (6,6,3,7) and (7,7,5,7)
Year: 2026
Type: preprint
Abstract: For integers m,n,k,l with k+l=m+n-2 and an integral zero P^{k+l}_{m-1}(k)=0 of a binary Krawtchouk polynomial, Heo and Kiem (Linear Algebra Appl. 567 (2019) 167-179, Question 1.2) asked whether there are subspaces D, E of C^m (x) C^n of codimensions k, l such that no nonzero x, y satisfy x(x)y in D and conj(x)(x)y in E. We give explicit pairs (D, E) for the open cases (5,5,3,5), (6,6,3,7) and (7,7,5,7) (hence also (5,5,5,3), (6,6,7,3) and (7,7,7,5)), each with a matrix-inequality certificate (as in arXiv:2611.08518, Sec. 5) verified in exact rational arithmetic. The release contains the note (PDF + LaTeX), the data, and a verification script (python scripts/verify_all.py; numpy + sympy; about 3-15 minutes). Work done with AI assistance (Anthropic Claude); reviewed by separate AI referees (reports included), not yet by a human expert.
Author: Fugsukjit, Vichien
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225817335
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