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Real algebraic geometry is the study of real solutions to algebraic equations with real coefficients. Its methods are rather different from classical algebraic geometry, which is typically done over an algebraically closed field (like the complex numbers).

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Tropical polynomial Positivstellensatz

http://logic.pdmi.ras.ru/~grigorev/pub/tropical_null.pdf Tropical dual and primary effective Nullstellensaetze http://logic.pdmi.ras.ru/~grigorev/pub/dim_np_completeness.pdf
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