Era | 1860 | 1900 | 1940 | 1980 | 2020 |
---|---|---|---|---|---|

Tensors are … | Lengths | Forms | Froms & Spaces | Grids, Forms, Spaces, & Multilinear maps | …all things distributive. |

Used by… | Hamilton | Geometry, Physics | Geometry, Kinematics, Relativity, Quantum Physics, Algebra, Analysis | _{Finite Geometry, Differential Geometry, Kinematics, Relativity, Quantum Mechanics, Representations, Algebras, PDEs, Categories, Model Theory, Informatics, Statistics} |
_{Finite Geometry, Differential Geometry, Kinematics, Relativity, Quantum Mechanics, Representations, Algebras, PDEs, Sensors, Categories, Morley Rank, Isomorphism, Informatics, Statistics, Big Data, Machine Learning, Compression, Data Structures, Clustering, Decompositions, Dimension Reduction, Image Processing, Synthizing Data, Scanning,…and all the uses of which we haven’t a clue!} |

\[f = \ell_1^d + \cdots + \ell_r^d\]

Definition.The Waring rank of a homogeneous polynomial $f \in C[x_1, \ldots, x_n]_d$ is the minimum $r$ such thatfor some linear forms $\ell_1, \ldots, \ell_r \in C[x_1, \ldots, x_n]_1$.

Let $p\geq 3$ be a prime. Let $\Lambda(n,p)$ be the set of alternating matrices over $\mathbb{F}_p$, where $A\in M(n,p)$ is alternating if for all $v\in\mathbb{F}_p^n$, $v^tAv=0$. Let $\mathbb{A}\in\Lambda(n,q)^m$ be an $m$-tuple of alternating matrices. We say two $m$-tuple of alternating matrices $\mathbb{A}_1$ and $\mathbb{A}_2$ *pseudo-isometric*, if there exists $T\in \text{GL}(n,p)$ such that $\langle T^t\mathbb{A}_1T\rangle=\langle\mathbb{A}_2\langle$ (as subspaces), where $\langle \mathbb{A} \rangle$ denotes the linear space spanned by alternating matrices in the $m$-tuple $\mathbb{A}$.