**Semi-Regularity of Pairs of Boolean Polynomials**

*Timothy J. Hodges and Hari R. Iyer*

**Abstract: **Semi-regular sequences over $\mathbb{F}_2$ are sequences of homogeneous elements of the algebra $
B^{(n)}=\mathbb{F}_2[X_1,...,X_n]/(X_1^2,...,X_n^2)
$, which have a given Hilbert series and can be thought of as having as few relations between them as possible. It is believed that most such systems are 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. We investigate the case where the sequence has length two and give an almost complete description of the number of semi-regular sequences for each $n$.

**Category / Keywords: **foundations / Semi-regular sequence

**Date: **received 18 Dec 2020, last revised 21 Dec 2020

**Contact author: **timothy hodges at uc edu

**Available format(s): **PDF | BibTeX Citation

**Version: **20201221:223644 (All versions of this report)

**Short URL: **ia.cr/2020/1585

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