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A note on extensions of multilinear maps defined on multilinear varieties

Part of: Additive number theory; partitions

Published online by Cambridge University Press:  30 April 2021

W. T. Gowers  and
L. Milićević
W. T. Gowers
Affiliation:
Collège de France and University of Cambridge, 11, Place Marcelin-Berthelot, Paris75231, France (wtg10@dpmms.cam.ac.uk)
L. Milićević
Affiliation:
Mathematical Institute of the Serbian Academy of Sciences and Arts, Kneza Mihaila 36, Belgrade11000, Serbia (luka.milicevic@turing.mi.sanu.ac.rs)
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Abstract

Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$. A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ 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 $c$ and all choices of $x_i \in G_i$, $i\not =c$, the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$. Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

Keywords

multilinear mapsmultilinear varietiesextensions of maps

MSC classification

Primary: 11P99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 2 , May 2021 , pp. 148 - 173
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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