Bivariate hermite interpolation and applications to Algebraic Geometry

Summary We try to solve the bivariate interpolation problem (1.3) for polynomials (1.1), whereS is a lower set of lattice points, and for theq-th interpolation knot,A _q is the set of orders of derivatives that appear in (1.3). The number of coefficients |S| is equal to the number of equations |A _q |. If this is possible for all knots in general position, the problem is almost always solvable (=a.a.s.). We seek to determine whether (1.3) is a.a.s. An algorithm is given which often gives a positive answer to this. It can be applied to the solution of a problem of Hirschowitz in Algebraic Geometry. We prove that for Hermite conditions (1.3) (when allA _q are lower triangles of orderp) andP is of total degreen, (1.3) is a.a.s. for allp=1, 2, 3 and alln, except for the two casesp=1,n=2 andp=1,n=4.Dedicated to R. S. Varga on the occasion of his sixtieth birthdayThis work has been partly supported by the Texas ARP and the Deutsche Forschungsgemeinschaft