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