WARING LENGTH TWO FOR THE SUPERINVOLUTIVE L’VOV–KAPLANSKY COUNTEREXAMPLE ON UPPER TRIANGULAR MATRICES

Source abstract

Abstract Let F $F$ upper F be an algebraically closed field of characteristic zero. Campedel, Fagundes and Ioppolo [‘Upper triangular matrices with superinvolution: identities and images of multilinear polynomials’, Bull. Braz. Math. Soc. (N.S.) 57 (2026), Article no. 27] recently established a qualitative break from the L’vov–Kaplansky conjecture by proving that the multilinear ∗ $*$ asterisk -polynomial f ( y + , z + ) = y + z + $f(y^+,z^+)=y^+z^+$ f left parenthesis y Superscript plus Baseline comma z Superscript plus Baseline right parenthesis equals y Superscript plus Baseline z Superscript plus has a nonlinear image on the upper triangular matrix algebra A n = UT n ( F ) $A_n=\mathrm { UT}_n(F)$ upper A Subscript n Baseline equals upper U upper T Subscript n Baseline left parenthesis upper F right parenthesis ( n ≥ 4 $n\geq 4$ n greater than or equals 4 ) under the alternating elementary Z 2 $\mathbb Z_2$ double struck upper Z 2 -grading and super-reflection superinvolution. We provide the strict quantitative counterpart to this phenomenon by determining the exact Waring length of the counterexample in the first nontrivial dimensions. For n = 4 $n=4$ n equals 4 and n = 5 $n=5$ n equals 5 , we prove that the linear span of the image is the entire odd component ( A n ) 1 $(A_n)_1$ left parenthesis upper A Subscript n Baseline right parenthesis Subscript 1 and that the additive defect is strictly minimal, with Waring length 2 $2$ 2 . Our proof is constructive: we compute the products ( A n ) 0 + ( A n ) 1 + $(A_n)_0^+(A_n)_1^+$ left parenthesis upper A Subscript n Baseline right parenthesis Subscript 0 Superscript plus Baseline left parenthesis upper A Subscript n Baseline right parenthesis Subscript 1 Superscript plus via the orbit structure of matrix units and provide explicit two-value decompositions for arbitrary odd upper triangular matrices.