Complexity Analysis of Random Massive Alge- braic System
نویسندگان
چکیده
We build a new kind of random polynomial model, Massive Algebraic System (MAS), which is equivalent to algebraic polynomial system in solvability. This model focused on Finite Fields, especially on Z2, is NPcomplete. By results of Population Dynamics, we find these systems undergo 1-Replica Symmetry Breaking (1RSB) and high order Replica Symmetry Breaking. This phenomenon indicates subtle structure and profound complexity of these systems. For determining the solvability of these random systems, Moments Method provides lower bounds α ∼ 0.5 and upper bounds α = 1 of threshold value. And through the dynamical analysis of Leaf-Removing algorithm, the lower bounds are improved by αl(p), for which αl(1) = 0.562 and αl(0) = 0.818. Well understanding of this brand-new random algebraic model, which brings on new ways of complexity variation, will be instructive to polynomial equations with massive variables. Mathematics Subject Classification (2000). Primary 68Q17; Secondary 93A15.
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