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In the case of univariate polynomial it is possible to (upper) bound the number of roots in the given (real) interval, see Descartes' rule of signs, Budan's theorem, Vincent's theorem, Sturm's theorem, ... Is there anything similar for a system of polynomials? Something like "Multivariate Descartes' Rule" by Itenberg & Roy, which unfortunately doesn't seem to work ("On Multivariate Descartes' Rule - A Counterexample" by Li & Wang)...

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