Algebraic and Semialgebraic Proofs: Methods and Paradoxes

The aim of the present paper is the following: - Examine critically some features of the usual algebraic proof protocols, in particular the "test phase" that checks if a theorem is "true" or not, depending on the existence of a non-degenerate component on which it is true; this form of "truth" leads to paradoxes, that are analyzed both for real and complex theorems. - Generalize these proof tools to theorems on the real field; the generalization relies on the construction of the real radical, and allows to consider inequalities in the statements. - Describe a tool that can be used to transform an algebraic proof valid for the complex field into a proof valid for the real field. - Describe a protocol, valid for both complex and real theorems, in which a statement is supplemented by an example; this protocol allows us to avoid most of the paradoxes.

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