A note on extensions of multilinear maps defined on multilinear varieties
W. T. Gowers, L. Mili\'cevi\'c

TL;DR
This paper proves that multilinear maps defined on certain algebraic subsets called multilinear varieties can be extended to multilinear maps on the entire product space, with the extension's complexity depending polynomially on the original codimension.
Contribution
It establishes a polynomial bound on extending multilinear maps from multilinear varieties to the entire product space.
Findings
Multilinear maps on multilinear varieties can be extended to the whole space.
The extension's codimension grows polynomially with the original codimension.
Provides a theoretical foundation for extending multilinear structures in finite fields.
Abstract
Let be finite-dimensional vector spaces over a finite field . A multilinear variety of codimension is a subset of defined as the zero set of forms, each of which is multilinear on some subset of the coordinates. A map defined on a multilinear variety is multilinear if for each coordinate and all choices of , , the restriction map is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension coincides on a multilinear variety of codimension with a multilinear map defined on the whole of .
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Taxonomy
TopicsCoding theory and cryptography · Finite Group Theory Research · Tensor decomposition and applications
