Complexity and real computation
Complexity and real computation
High probability analysis of the condition number of sparse polynomial systems
Theoretical Computer Science - Algebraic and numerical algorithm
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We consider a random polynomial system with m equations and m real unknowns. Assume all equations have the same degree d and the law on the coefficients satisfies the Kostlan-Shub-Smale hypotheses. It is known that E(N^X)=d^m^/^2 where N^X denotes the number of roots of the system. Under the condition that d does not grow very fast, we prove that limsup"m"-"+"~VarN^Xd^m^/^2==3 then VarN^Xd^m^/^2-0 as m-+~, which implies N^Xd^m^/^2-1 in probability.