Scalar multiplication complexity of iterated commutators

Determine the minimum number of scalar multiplications required to compute the left-normed iterated commutator [A_1,\ldots,A_k] of k arbitrary 2 × 2 matrices for every k ≥ 3.

Background

The paper considers the computational complexity of evaluating the left-normed iterated commutator of k 2 × 2 matrices. Although the algebraic geometry of the corresponding vanishing locus is completely resolved in the paper, the optimal arithmetic complexity of computing the commutator remains unresolved for iterated commutators of length at least three.

References

a question in algebraic complexity theory -- how many scalar multiplications are needed to compute \left\lbrack A_{1},\ldots,A_{k} \right\rbrack? -- which remains genuinely open for $k \geq 3$ and which the author is presently investigating.

— The coordinate ring of the k-fold iterated commutator locus for 2x2 matrices  (2609.05386 - Snellman, 4 Sep 2026) in Section 1, Introduction