Jeyoung Song | Searcharxiv

arXiv2026

The rank of the $5\times 5$ permanent tensor is sixteen

In this paper, we show that the tensor rank of the $5\times 5$ permanent tensor is exactly $16$ over fields of characteristic zero, by proving the lower bound matching the known upper bound. The $5\times 5$ permanent tensor is a symmetric tensor corresponding to the monomial $x_1x_2x_3x_4x_5$, whose Waring rank is known to be $16$. Previously, it was known, by the higher-order Koszul flattening and Fischer's formula, that its tensor rank is either $15$ or $16$.

math.AG↗

arXiv2025

On the construction of a counterexample to Strassen's rank additivity conjecture

The rank additivity conjecture, first formulated by Volker Strassen in 1973, states that the rank of the direct sum of two independent tensors is equal to the sum of their individual ranks. In the last decades, this conjecture has been a central topic in tensor rank theory and its implications for computational complexity. In 2019, Yaroslav Shitov disproved this conjecture in its general form by showing the existence of a counter-example using a dimension counting argument. In this paper, we provide an overview of the Strassen problem and Shitov's work and revisit his counterexample with a detailed explanation, offering an alternative proof.

math.AG↗