Interpolation in algebraic geometry

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Here is a typical example of an interpolation problem: Given a collection of data (t1,x1) (t2,x2) (t3,x3) .... (tr,xr), find a polynomial function x=f(t) fitting the data, i.e., xi=f(ti) i=1,...,r. Similarly, we could prescribe the first N derivatives of f(t) at t1,....,tr. Our main focus is constrained interpolation problems, where the data and the interpolating function satisfy additional polynomial equations. For instance, given data (t1,x1,y1) (t2,x2,y2) ...... satisfying a constraint C(t,x,y)=0, find rational functions x=f(t) y=g(t) so that xi=f(ti) yi=g(ti) C(t,f(t),g(t))=0. Generally, constrained interpolation prolems need not have solutions; we shall discuss geometric conditions on the constraints where solutions are possible.