Generalized Mixed Volume of Polynomial Systems
summary
Defines a generalized mixed volume computed by introducing replacement variables for repeated subexpressions and then randomizing coefficients on the fixed algebraic set; shows examples where the BKK mixed volume 4 drops to actual root count 2, and a serial link robot whose BKK count 288 collapses to 16 under an appropriate replacement scheme. Uses polyhedral homotopy reasoning, Bertini randomization and explicit replacement constructions, notes GMV can equal true roots, enlarges the system and extends to positive‑dimensional varieties.
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