Homological characterization of bounded F2-regularity

[摘要] Semi-regular sequences over F-2 are sequences of homogeneous elements of the algebra B-(n) = F-2[X-1, ..., X-n]/(X-1(2), ..., X-n(2)), which have as few relations between them as possible. It is believed that most such systems are F-2-semi-regular and this property has important consequences for understanding the complexity of Grobner basis algorithms such as F4 and F5 for solving such systems. In fact even in one of the simplest and most important cases, that of quadratic sequences of length n in n variables, the question of the existence of semi-regular sequences for all n remains open. In this paper we present a new framework for the concept of F-2-semi-regularity which we hope will allow the use of ideas and machinery from homological algebra to be applied to this interesting and important open question. First we introduce an analog of the Koszul complex and show that F-2-semi-regularity can be characterized by the exactness of this complex. We show how the well known formula for the Hilbert series of a F-2-semi-regular sequence can be deduced from the Koszul complex. Finally we show that the concept of first fall degree also has a natural description in terms of the Koszul complex. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-12-15

[关键词] Semi-regularity;Finite field;Koszul complex