1 Introduction
2 Notations and terminology
3 Matrix F5
4 Gröbner bases with signatures
4.2 Signature Gröbner bases
5 Generic signature Gröbner basis computation
6 S-Pair Elimination
6.1 Eliminating S-pairs by known syzygies
6.2 Uniqueness of S-pairs at a given signature
7 Rewrite bases
7.1 Combining elimination criteria
7.2 An algorithm computing rewrite bases
7.3 Choosing a rewrite order
8.2 F5C – Improved S-pair generation
8.3 Extended F5 criteria
9 Exploiting algebraic structures
9.1 F5/2 – Improved computations over F2
9.2 An F5 variant for bihomogeneous ideals generated by polynomials of bidegree (1,1)
9.3 An F5 variant for SAGBI Gröbner bases
10 F5 and the quest of termination
10.1 Proving F5's termination
10.2 Variants of F5 to ensure termination algorithmically
11 Signature-based Gröbner basis algorithms using ⊴rat
11.1 Arri and Perry's work – AP
11.2 The TRB algorithm – top reductional basis
11.3 The GBGC algorithm – generalized criteria
11.4 The G2V algorithm
11.5 The GVW algorithm
11.6 The SB algorithm
11.7 The SSG algorithm
12 Using Buchberger's criteria in signature-based Gröbner basis algorithms
12.1 Buchberger's criteria
12.3 GVW's 2013 revision
13 s-reductions using linear algebra
14 Experimental results
14.3 Observations
15 Concluding remarks