18.S997 Fall 2012 The Polynomial Method: Degree Reduction
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چکیده
(should we prove it?...) If deg(S) is significantly smaller than |S|, then it means that S has more algebraic structure than a generic set. We are going to explore the connection between combinatorial properties of a set S and its algebraic structure. We will see that interesting examples in the kind of incidence geometry questions we have been studying need to have algebraic structure. Once we prove that a set has some algebraic structure, it makes sense to try to use that structure to study the set. As a warmup, we consider a set of L lines in 3 F . It’s easy to find a degree L polynomial that vanishes on the L lines, but in fact we can do better.
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