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Most Hard Equations are Easy
While randomization has long been used profitably in optimization and computer science, it's importance in algebraic geometry has only recently been realized. In particular, the study of random systems of equations leads to some beautiful interactions between geometry, applications, and algorithmic complexity.
We survey some of these developments from the point of view of polynomial system solving, focusing particularly on real (as opposed to complex) solutions. We will also see how centuries-old questions on counting real solutions can be solved with modern ideas from algebraic geometry (and a little randomization).
No background in algebraic geometry or algorithms is assumed.
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