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Counting Zeros over Finite Fields with Grobner Bases May 17, 2009
 

Summary: Counting Zeros over Finite Fields with Gršobner Bases
Sicun Gao
May 17, 2009
Contents
1 Introduction 2
2 Finite Fields, Nullstellensatz and Gršobner Bases 5
2.1 Ideals, Varieties and Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Gršobner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Hilbert's Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Counting with Gršobner Bases 24
3.1 Nullstellensatz in Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 |SM(J + Żxq
- Żx )| = |V (J)| . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Algorithm Analysis 29
4.1 Analysis of Buchberger's Algorithm . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Counting Standard Monomials . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5 A Practical #SAT Solver 34
5.1 DPLL-based Approaches to #SAT . . . . . . . . . . . . . . . . . . . . . . . 34
5.2 Gršobner Bases in Boolean Rings . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

  

Source: Avigad, Jeremy - Departments of Mathematical Sciences & Philosophy, Carnegie Mellon University
Kalla, Priyank - Department of Electrical and Computer Engineering, University of Utah

 

Collections: Engineering; Mathematics; Multidisciplinary Databases and Resources