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Summary: Counting Zeros over Finite Fields Using
Gršobner Bases
Sicun Gao
May, 2009
Contents
1 Introduction 2
2 Finite Fields, Nullstellensatz and Gršobner Bases 6
2.1 Ideals, Varieties and Finite Fields . . . . . . . . . . . . . . . . 6
2.2 Gršobner Bases . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Hilbert's Nullstellensatz . . . . . . . . . . . . . . . . . . . . . 21
3 Counting with Gršobner Bases 26
3.1 Nullstellensatz in Finite Fields . . . . . . . . . . . . . . . . . 26
3.2 |SM(J + Żxq - Żx )| = |V (J)| . . . . . . . . . . . . . . . . . . 28
4 Algorithm Analysis 32
4.1 Analysis of Buchberger's Algorithm . . . . . . . . . . . . . . . 32
4.2 Counting Standard Monomials . . . . . . . . . . . . . . . . . 35
5 A Practical #SAT Solver 37
5.1 DPLL-based Approaches to #SAT . . . . . . . . . . . . . . . 37
5.2 Gršobner Bases in Boolean Rings . . . . . . . . . . . . . . . . 39
5.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 40
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