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Geometric aspects of polynomial interpolation in more variables and of Waring’s problem European
 Congress of Mathematics, Vol. I (Barcelona
, 2001
"... Abstract. In this paper I treat the problem of determining the dimension of the vector space of homogeneous polynomials in a given number of variables vanishing with some of their derivatives at a finite set of general points in projective space. I will illustrate the geometric meaning of this probl ..."
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Abstract. In this paper I treat the problem of determining the dimension of the vector space of homogeneous polynomials in a given number of variables vanishing with some of their derivatives at a finite set of general points in projective space. I will illustrate the geometric meaning of this problem and the main results and conjectures about it. I will finally point out its connection with the socalled Waring’s problem for forms, of which I will also indicate the geometric meaning. 1.
Linear Systems of Plane Curves
 Notices of the AMS
, 1999
"... Interpolation with polynomials is a subject that has occupied mathematicians ’ minds for millenia. The general problem can be informally phrased as: Given a set of points {(xi,yi)} in the plane, find a ..."
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Cited by 29 (2 self)
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Interpolation with polynomials is a subject that has occupied mathematicians ’ minds for millenia. The general problem can be informally phrased as: Given a set of points {(xi,yi)} in the plane, find a
Problems and Progress: A survey on fat points in P²
, 2001
"... This paper, which expands on a talk given at the International Workshop on Fat ..."
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Cited by 16 (2 self)
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This paper, which expands on a talk given at the International Workshop on Fat
Linear systems with multiple base points
 in P 2 , Adv. Geom
"... Abstract: Given positive integers m1, m2,..., mn, and t, and n points of P 2 in general position, consider the linear system of curves of degree t which have multiplicity at least mi at the ith point. In this paper bounds are given for the minimal degree such that the linear system is nonempty and ..."
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Cited by 13 (8 self)
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Abstract: Given positive integers m1, m2,..., mn, and t, and n points of P 2 in general position, consider the linear system of curves of degree t which have multiplicity at least mi at the ith point. In this paper bounds are given for the minimal degree such that the linear system is nonempty and for the minimal degree such that it is regular, often improving substantially what was previously known. As an application, for certain m and n (both of which can be large) the Hilbert function and minimal free resolution are determined for symbolic powers I (m) for the ideal I defining n general points of P 2; very few cases were known previously with both m and n large. We work over an algebraically closed field of characteristic zero. I.
Free Resolutions of Fat Point Ideals on P 2
 Journal of Pure and Applied Algebra
, 1998
"... Minimal free resolutions for homogeneous ideals corresponding to certain 0dimensional subschemes of P 2 defined by sheaves of complete ideals are determined implicitly. All work is over an algebraically closed field of arbitrary characteristic. I. ..."
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Cited by 9 (1 self)
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Minimal free resolutions for homogeneous ideals corresponding to certain 0dimensional subschemes of P 2 defined by sheaves of complete ideals are determined implicitly. All work is over an algebraically closed field of arbitrary characteristic. I.
Orecchia: Bivariate Hermite interpolation and linear systems of plane curves with base fat points
 Computer mathematics, Lecture Notes Ser. Comput., 10, World Sci. Publishing, River Edge, NJ, (2003) 87–102. of the Veronese and Applications 797
"... It is still an open question to determine in general the dimension of the vector space of bivariate polynomials of degree at most d which have all partial derivatives up through order mi − 1 vanish at each point pi (i = 1,..., n), for some fixed integer mi called multiplicity at pi. When the multip ..."
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It is still an open question to determine in general the dimension of the vector space of bivariate polynomials of degree at most d which have all partial derivatives up through order mi − 1 vanish at each point pi (i = 1,..., n), for some fixed integer mi called multiplicity at pi. When the multiplicities are all equal, to m say, this problem has been attacked by a number of authors (Lorentz and Lorentz, Ciliberto and Miranda, Hirschowitz) and there are a number of good conjectures (Hirschowitz, Ciliberto and Miranda) on the dimension of these interpolating spaces. The determination of the dimension has been already solved for m ≤ 12 and all d and n by a degeneration technique and some ad hoc geometric arguments. Here this technique is applied up through m = 20; since it fails in some cases, we resort (in these exceptional cases) to the bivariete Hermite interpolation with the support of a simple idea suggested by Gröbner bases computation. In summary we are able to prove that the dimension of the vector space is the expected one for 13 ≤ m ≤ 20. 1.
On Nagata’s Conjecture
 J. Alg
"... Abstract: This paper gives an improved lower bound on the degrees d such that for general points p1,..., pn ∈ P 2 and m> 0 there is a plane curve of degree d vanishing at each pi with multiplicity at least m. In this paper we work over an arbitrary algebraically closed field. For positive integer ..."
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Abstract: This paper gives an improved lower bound on the degrees d such that for general points p1,..., pn ∈ P 2 and m> 0 there is a plane curve of degree d vanishing at each pi with multiplicity at least m. In this paper we work over an arbitrary algebraically closed field. For positive integers m and n, define d(m, n) to be the least integer d such that for general points p1,..., pn ∈ P2 there is a curve of degree d vanishing at each point pi with multiplicity at least m. For n ≥ 10, Nagata [N1] conjectures that d(m, n)> m √ n, and proves this when n> 9 is a square. (For n ≤ 9, applying methods of [N2] it can be
systems in P 2 with base points of bounded multiplicity
 Department of Mathematics, Harvard University
"... Abstract. We present a proof of the HarbourneHirschowitz conjecture for linear systems with multiple points of order 7 or less. This uses a wellknown degeneration of the plane developed by Ciliberto and Miranda as well as a combinatorial game that arises from specializing points onto lines. Conten ..."
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Abstract. We present a proof of the HarbourneHirschowitz conjecture for linear systems with multiple points of order 7 or less. This uses a wellknown degeneration of the plane developed by Ciliberto and Miranda as well as a combinatorial game that arises from specializing points onto lines. Contents
Classifying Hilbert functions of fat point subschemes in P²
, 2008
"... The paper [GMS] raised the question of what the possible Hilbert functions are for fat point subschemes of the form 2p1 + · · · + 2pr, for all possible choices of r distinct points in P 2. We study this problem for r points in P 2 over an algebraically closed field k of arbitrary characteristic ..."
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The paper [GMS] raised the question of what the possible Hilbert functions are for fat point subschemes of the form 2p1 + · · · + 2pr, for all possible choices of r distinct points in P 2. We study this problem for r points in P 2 over an algebraically closed field k of arbitrary characteristic in case either r ≤ 8 or the points lie on a (possibly reducible) conic. In either case, it follows from [H2] and [H3] that there are only finitely many configuration types of points, where our notion of configuration type is a generalization of the notion of a representable combinatorial geometry, also known as a representable simple matroid. (We say p1,..., pr and p ′ 1,..., p ′ r have the same configuration type if for all choices of nonnegative integers mi, Z = m1p1 + · · · + mrpr and Z ′ = m1p ′ 1 + · · · + mrp ′ r have the same Hilbert function.) Assuming either that 7 ≤ r ≤ 8 (see [GuH] for the cases r ≤ 6) or that the points pi lie on a conic, we explicitly determine all the configuration types, and show how the configuration type and the coefficients mi determine (in an explicitly computable way) the Hilbert function (and sometimes the graded Betti numbers) of Z = m1p1 + · · · + mrpr. We demonstrate our results by explicitly listing all Hilbert functions for schemes of r ≤ 8 double points, and for each Hilbert function we state precisely how the points must be arranged (in terms of the configuration type) to obtain that Hilbert function.