On the existence of homogeneous semi-regular sequences in F2 [X1,..., Xn]/(X12 ,..., Xn2 )

Semi-regular sequences over F-2 are sequences of homogeneous elements of the algebra B-(n) = F-2 [X-1, X-n]/ (X?,..., X!), which have as few relations between them as possible. They were introduced in order to assess the complexity of Grobner basis algorithms such as F-4 and F-5 for the solution of polynomial equations. Despite the experimental evidence that semi-regular sequences are common, it was unknown whether there existed semi-regular sequences for all n, except in extremely trivial situations. We prove some results on the existence and non-existence of semi-regular sequences. In particular, we show that if an element of degree d in B-(n) is semi-regular, then we must have n <= 3d. Also, we show that if d = 2(t) and n = 3d, then there exists a semi-regular element of degree d establishing that the bound is sharp for infinitely many n. Finally, we generalise the result of nonexistence of semi-regular elements to the case of sequences of a fixed length in. (C) 2017 Elsevier Inc. All rights reserved.