Related papers: A New Formula for the Determinant and Bounds on It…

In this paper, we present a new formula of the determinant tensor $det_n$ for $n \times n$ matrices. In \cite{kim2023newdet4}, Kim, Ju, and Kim found a new formula of $4 \times 4$ determinant tensor $det_4$ which is available when the base…

We show that the border rank of the $4 \times 4$ determinant tensor is at least $12$ over $\mathbb{C}$, using the fixed ideal theorem introduced by Buczy\'nska-Buczy\'nski and the method by Conner-Harper-Landsberg. Together with the known…

We prove a lower bound on the rank of tensors constructed from families of linear maps that `expand' the dimension of every subspace. Such families, called {\em dimension expanders} have been studied for many years with several known…

In this paper, we present a new formula for the determinant of a $4 \times 4$ matrix. We approach via the sparse optimization problem and derive the formula through the Least Absolute Shrinkage and Selection Operator (LASSO). Our formula…

The results of Strassen and Raz show that good enough tensor rank lower bounds have implications for algebraic circuit/formula lower bounds. We explore tensor rank lower and upper bounds, focusing on explicit tensors. For odd d, we…

We develop a notion of {\em inner rank} as a tool for obtaining lower bounds on the rank of matrix multiplication tensors. We use it to give a short proof that the border rank (and therefore rank) of the tensor associated with $n\times n$…

Let $M_{\langle u,v,w\rangle}\in C^{uv}\otimes C^{vw}\otimes C^{wu}$ denote the matrix multiplication tensor (and write $M_n=M_{\langle n,n,n\rangle}$) and let $det_3\in ( C^9)^{\otimes 3}$ denote the determinant polynomial considered as a…

The Waring rank of the generic $d \times d$ determinant is bounded above by $d \cdot d!$. This improves previous upper bounds, which were of the form an exponential times the factorial. Our upper bound comes from an explicit power sum…

We provide a nontrivial bound on the rank of any tensor $T$ over the quaternions $\mathbb{H}$ in the $n_1\times n_2\times n_3$ cases where $2\leq n_i\leq 3$. We describe a decomposition of $T$ into $3$ simple tensors in the $2\times 2\times…

Matrices of rank at most k are defined by the vanishing of polynomials of degree k + 1 in their entries (namely, their (k + 1)-times-(k + 1)-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this…

The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P =…

We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric…

We study an upper bound of ranks of $n$-tensors with size $2\times\cdots\times2$ over the complex and real number field. We characterize a $2\times 2\times 2$ tensor with rank 3 by using the Cayley's hyperdeterminant and some function. Then…

We study the tensor rank of the tensor corresponding to the algebra of n-variate complex polynomials modulo the dth power of each variable. As a result we find a sequence of tensors with a large gap between rank and border rank, and thus a…

We investigate new lower bounds on the tensor rank of the determinant and the permanent tensors via recursive usage of the Koszul flattening method introduced by Landsberg-Ottaviani and Hauenstein-Oeding-Ottaviani-Sommese. Our lower bounds…

For tensors of fixed order, we establish three types of upper bounds for the geometric rank in terms of the subrank. Firstly, we prove that, under a mild condition on the characteristic of the base field, the geometric rank of a tensor is…

We study the subrank of real order-three tensors and give an upper bound to the subrank of a real tensor given its complex subrank. Using similar arguments to those used by Bernardi-Blekherman-Ottaviani, we show that all subranks between…

The tensor rank and border rank of the $3 \times 3$ determinant tensor is known to be $5$ if characteristic is not two. In this paper, we show that the tensor rank remains $5$ for fields of characteristic two as well. We also include an…

We study tensors in $\C^{m\times n\times l}$ whose border rank is $l$. We characterize the tensors in $\C^{3\times 3\times 4}$ and in $\C^{4\times 4\times 4}$ of border rank 4 at most.

This paper is devoted to the study of lower and upper bounds for the number of vertices of the polytope of $n\times n\times n$ stochastic tensors (i.e., triply stochastic arrays of dimension $n$). By using known results on polytopes (i.e.,…