Proofs by example
Abstract
We study the proof scheme "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 article addresses the problem of constructing 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 method is based on diophantine geometry, in particular an arithmetic Bézout theorem, an arithmetic Nullstellensatz, and a new effective LiouvilleLojasiewicz type inequality for algebraic varieties. As an application we discuss theorems from plane geometry and how to prove them by example.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.00480
 Bibcode:
 2019arXiv190900480M
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  Logic;
 03F03;
 03F07;
 11G35;
 13P10;
 14Q20
 EPrint:
 47 pages, 1 figure