Let $F$ 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 $*$-polynomial $f(y^+,z^+)=y^+z^+$ has a nonlinear image on the upper triangular matrix algebra $A_n=\mathrm { UT}_n(F)$ ($n\geq 4$) under the alternating elementary $\mathbb Z_2$-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 $n=4$ and $n=5$, we prove that the linear span of the image is the entire odd component $(A_n)_1$ and that the additive defect is strictly minimal, with Waring length $2$. Our proof is constructive: we compute the products $(A_n)_0^+(A_n)_1^+$ via the orbit structure of matrix units and provide explicit two-value decompositions for arbitrary odd upper triangular matrices.

We study the regularity of the higher secant varieties of ${{\mathbb{P}}^{1}}\times {{\mathbb{P}}^{n}}$ , embedded with divisors of type $\text{(}d\text{,}\,\text{2)}$ and $(d,3)$ . We produce, for the highest defective cases, a “determinantal” equation of the secant variety. As a corollary, we prove that the Veronese triple embedding of ${{\mathbb{P}}^{n}}$ is not Grassmann defective.