A universal sequence of tensors for the asymptotic rank conjecture

The exponent $σ(T)$ of a tensor $T\in\mathbb{F}^d\otimes\mathbb{F}^d\otimes\mathbb{F}^d$ over a field $\mathbb{F}$ captures the base of the exponential growth rate of the tensor rank of $T$ under Kronecker powers. Tensor exponents are fundamental from the standpoint of algorithms and computational complexity theory; for example, the exponent $ω$ of matrix multiplication can be characterized as $ω=2σ(\mathrm{MM}_2)$, where $\mathrm{MM}_2\in\mathbb{F}^4\otimes\mathbb{F}^4\otimes\mathbb{F}^4$ is the tensor that represents $2\times 2$ matrix multiplication. Our main result is an explicit construction of a sequence $\mathcal{U}_d$ of zero-one-valued tensors that is universal for the worst-case tensor exponent; more precisely, we show that $σ(\mathcal{U}_d)=σ(d)$ where $σ(d)=\sup_{T\in\mathbb{F}^d\otimes\mathbb{F}^d\otimes\mathbb{F}^d}σ(T)$. We also supply an explicit universal sequence $\mathcal{U}_Δ$ localised to capture the worst-case exponent $σ(Δ)$ of tensors with support contained in $Δ\subseteq [d]\times[d]\times [d]$; by combining such sequences, we obtain a universal sequence $\mathcal{T}_d$ such that $σ(\mathcal{T}_d)=1$ holds if and only if Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)] holds for $d$. Finally, we show that the limit $\lim_{d\rightarrow\infty}σ(d)$ exists and can be captured as $\lim_{d\rightarrow\infty} σ(D_d)$ for an explicit sequence $(D_d)_{d=1}^\infty$ of tensors obtained by diagonalisation of the sequences $\mathcal{U}_d$. As our second result we relate the absence of polynomials of fixed degree vanishing on tensors of low rank, or more generally asymptotic rank, with upper bounds on the exponent $σ(d)$. Using this technique, one may bound asymptotic rank for all tensors of a given format, knowing enough specific tensors of low asymptotic rank.