The L'vov–Kaplansky Conjecture for Standard Polynomials

The L’vov–Kaplansky conjecture asks whether the image of a multilinear polynomial on a full matrix algebra is a vector subspace. This preprint determines the sets of individual values attained by every unnormalised standard polynomial over algebraically closed fields of characteristic zero, in all degrees and matrix orders. For odd degree 2r − 1, the image is zero when the matrix order n is less than r, and the full matrix algebra when n is at least r. For even degree 2r, the image is zero when n is at most r, and the space of traceless matrices when n is at least r + 1. The odd-degree proof uses explicit linear slices and Jordan forms. The even-degree argument combines a uniform nilpotent construction and additive transfer for admissible cores with generic-nullity results of Brassil–Reichstein and Lin–Tang–Zhao, extending a method of Qiyue Tang to obtain actual polynomial values. The accompanying source archive contains the complete LaTeX manuscript, five Python verification programs, nine saved certificates, reproducibility instructions and licensing information. GitHub repository: https://github.com/oddcurios235/lvov-kaplansky-standard-polynomials