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1 results

  • 5 - The Joints Problem and Degree Reduction

  • Adam Sheffer, Bernard M. Baruch College, City University of New York
  • Book:
    Polynomial Methods and Incidence Theory
    Published online:
    17 March 2022
    Print publication:
    24 March 2022, pp 66-75

  • Summary

    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.