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Published online by Cambridge University Press: 17 March 2022
Before their seminal distinct distances paper, Guth and Katz wrote another paper that introduced a new polynomial method. In this chapter, we study one of the two problems that were resolved in that paper: the joints problem. The solution to this problem relies on a simple polynomial technique, which is based on polynomial interpolation. This is also a good warm-up for working in spaces of dimension larger than two.
We use the polynomial interpolation technique to study two additional problems. First, we study the sets in R^3 that are formed by the union of all lines that intersect three pairwise-skew lines. We then use the degree reduction technique to study polynomial interpolation of lines.
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