Expected term bases for generic multivariate Hermite interpolation Authors Marcin Dumnicki Institute of Mathematics Jagiellonian University Article

First Online: 06 July 2007 Received: 18 October 2004 Revised: 09 May 2007 DOI :
10.1007/s00200-007-0049-6

Cite this article as: Dumnicki, M. AAECC (2007) 18: 467. doi:10.1007/s00200-007-0049-6
Abstract The main goal of the paper is to find an effective estimation for the minimal number of points in \({\mathbb{K}}^{2}\) in general position for which the basis for Hermite interpolation consists of the first ℓ terms (with respect to total degree ordering). As a result we prove that the space of plane curves of degree at most d having singularities of multiplicity ≤ m in general position has the expected dimension if the number of low order singularities (of multiplicity k ≤ 12) is greater then some r (m , k ). Additionally, the upper bounds for r (m , k ) are given.

Keywords Multivariate interpolation Algebraic curves

References 1.

Alexander J. and Hirschowitz A. (2000). An asymptotic vanishing theorem for generic unions of multiple points.

Invent. Math. 140: 303–325

MATH CrossRef MathSciNet 2.

Alexander J. and Hirschowitz A. (1995). Polynomial interpolation in several variables.

J. Algebr. Geom. 4: 201–222

MATH MathSciNet 3.

Apel J., Stückrad J., Tworzewski P. and Winiarski T. (1999). Term bases for multivariate interpolation of Hermite type. Univ. Iagell. Acta Math. 37: 37–49

4.

Becker T. and Weispfenning V. (1993). Gröbner Bases. Springer, New York

MATH 5.

Cerlienco L., Mureddu M. (1995) From algebraic sets to monomial linear bases by means of combinatorial algorithms. Discr. Math. 139, 73–87

MATH CrossRef MathSciNet 6.

Ciliberto C. and Miranda R. (2000). Linear systems of plane curves with base points of equal multiplicity.

Trans. Am. Math. Soc. 352: 4037–4050

MATH CrossRef MathSciNet 7.

Fulton, W.: Algebraic Curves. W. A. Benjamin, Inc. (1978)

8.

Mignon T. (1998). Systèmes linéaires de courbes planes à singularités ordinaires imposées.

CRAS 327: 651–654

MATH MathSciNet 9.

Möller H.M. and Sauer T. (2000). H-bases for polynomial interpolation and system solving.

Adv. Comput. Math. 12: 335–362

MATH CrossRef MathSciNet 10.

Sauer, T.: Polynomial interpolation of minimal degree and Gröbner bases. London Math. Soc. Lecture Notes Ser. 251. Cambridge University Press, Cambridge (1998)