A note on extensions of multilinear maps defined on multilinear varieties

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.