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Interpolating Polynomial Equations and the complexity of an algebraic varietyColloquium
|Speaker: ||David Eisenbud, MSRI|
|Location: ||1147 MSB|
|Start time: ||Fri, May 12 2006, 1:10PM|
Given d distinct points on a line, elementary linear algebra shows that one can find a polynomial function of degree d-1 taking specified values at each of the points --- this is called interpolation. Interpolation on d points cannot be done with polynomials of degree d-2. On the other hand, polynomials of degree d-1 suffice even if the points are not on a line, but in a plane or higher dimensional space.
However, for "most" sets of d points in a higher-dimensional space, polynomials of much smaller degree suffice for interpolation. There is a remarkable formula giving the exact degree required for interpolation on any set of points, and it is related to deep questions about complexity in commutative algebra and algebraic geometry. I'll explain the ideas necessary to understand this formula, and talk about some questions to which it leads, including recent work of mine with Craig Huneke and Joe Harris.