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Proofs by example and numerical Nullstellensätze

Benjamin Matschke

Friday, 2021-12-03 10:30

We study the proof method "proof by example" in which a general statement can be proved by verifying it for a single example. This strategy can indeed work if the statement in question is an algebraic identity and the example is "generic". This talk addresses the problem of construction a practical example, which is sufficiently generic, for which the statement can be verified efficiently, and which even allows for a numerical margin of error. Our answer comes in the form of a numerical Nullstellensatz, which is based on Diophantine geomery, in particular an arithemetic Nullstellensatz and a new effective Liouville-Lojasiewicz type inequality.

If time permits we moreover consider "proofs by several examples", which in addition requires a conceptual notion of sufficient genericity of a set of points. Besides theoretical and algorithmic criteria for sufficient genericity, we obtain several new types of Nullstellensätze in the spirit of the combinatorial Nullstellensatz and the Schwartz-Zippel lemma, also for varieties.