|Authors:||Gowers, W. T.
Mathematical Institute of the Serbian Academy of Sciences and Arts
|Title:||A note on extensions of multilinear maps defined on multilinear varieties||Journal:||Proceedings of the Edinburgh Mathematical Society||Volume:||64||Issue:||2||First page:||148||Last page:||173||Issue Date:||2021||Rank:||M22||ISSN:||0013-0915||DOI:||10.1017/S0013091521000055||Abstract:||
Let be finite-dimensional vector spaces over a prime field. A multilinear variety of codimension at most 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 at most coincides on a multilinear variety of codimension with a multilinear map defined on the whole of. Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.
|Keywords:||extensions of maps | multilinear maps | multilinear varieties||Publisher:||Cambridge University Press|
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