On Randomized Semi-algebraic Test Complexity

AbstractWe investigate the impact of randomization on the complexity of deciding membership in a (semi-)algebraic subset X ⊂ Rm. Examples are exhibited where allowing for a certain error probability ϵ in the answer of the algorithms the complexity of decision problems decreases. A randomized (Ωk, {=, ≤})-decision tree (k ⊆ R a subfield) over m will be defined as a pair (T, μ) where μ a probability measure on some Rn and T is a (Ωk, {=, ≤})-decision tree over m + n. We prove a general lower bound on the randomized decision complexity for testing membership in an irreducible algebraic subset X ⊂ Rm and apply it to k-generic complete intersection of polynomials of the same degree, extending results in P. Bürgisser, T. Lickteig, and M.. Shub ((1992), J. Complexity8, 203-215) and P. Bürgisser ((1992), Research Report No. 8578-CS, Univ. Bonn). We also give applications to nongeneric cases, such as graphs of elementary symmetric functions, SL(m, R), and determinant varieties, extending results in (Lickteig, T. (1990), Tech. Rep. TR-90-90-052 Int. Comp. Science Inst., Berkeley, and Univ. Tübingen, Habilitationsschrift). The results are based on the methods from that paper and P. Bürgisser and T. Lickteig (1992), J. Pure Appl. Algebra81, 247-267)