Counting Zeros over Finite Fields Using Grobner Bases 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 Collections: Multidisciplinary Databases and Resources; Mathematics