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A note on extensions of multilinear maps defined on multilinear varieties | | W. T. Gowers
; L. Milićević
; | | Date: |
11 Jun 2019 | | Abstract: | Let $G_1, dots, G_k$ be finite-dimensional vector spaces over a finite field
$mathbb{F}$. A multilinear variety of codimension $d$ is a subset of $G_1
imes dots imes G_k$ defined as the zero set of $d$ forms, each of which is
multilinear on some subset of the coordinates. A map $phi$ defined on a
multilinear variety $B$ is multilinear if for each coordinate $d$ and all
choices of $x_i in G_i$, $i
ot=d$, the restriction map $y mapsto phi(x_1,
dots, x_{d-1}, y, x_{d+1}, dots, x_k)$ is linear where defined. In this note,
we show that a multilinear map defined on a multilinear variety of codimension
$d$ coincides on a multilinear variety of codimension $d^{O(1)}$ with a
multilinear map defined on the whole of $G_1 imesdots imes G_k$. | | Source: | arXiv, 1906.4807 | | Services: | Forum | Review | PDF | Favorites | |
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