Alexandru Dimca - UCA, LJAD - Keywords: interpolation, polynomial

Alexandru Dimca - UCA, LJAD - Keywords: interpolation, polynomial

Contribution title: Polynomial interpolation in higher dimensions

Polynomials can be used to approximate complicated curves and surfaces, for example, the shapes of letters in typography or the shape of a plane wing, given a few points. Such polynomials approximate in fact complicated functions of one or several variables, and having a bound on their degrees is essential in all the practical applications.

For the case of 1-dimensional interpolation, used to construct curves in the plane, one starts with a set of d data points, which are pairs (x,y) of real numbers, and looks for a polynomial P=P(t) of degree D=d-1 such that
P(x)=y for any given pair (x,y). Such a polynomial P exists and is unique, and P is called the Lagrange polynomial.

For the case of 2-dimensional interpolation, used to construct surfaces in a 3d space, the first entry x in the pair (x,y) is now a point in the plane. If we denote by X the set of all such points x, and assume again that we have d points in X, it is a difficult and challenging problem to find the minimal degree D such that there is a polynomial in two variables P=P(t1,t2) satisfying P(x)=y for all the points in X.
This degree depends on the geometry of the set of points X, as reflected in the Cayley-Bacharach Theorem and the
Segre-Harbourne-Gimigliano-Hirschowitz Conjecture.

A recent variant of this 2-dimensional problem consists in looking for polynomials P as above, having in addition a given Taylor expansion of a given order at one point z not in X, see [2]. The practical interest of this variant is that, for instance, a zero Taylor expansion of high order means very small values for the polynomial P in the neighborhood of the given point z.

To have a concrete example, one can consider the case when the polynomial P models the intensity of the electromagnetic field generated by some mobile phone antennas located at the given points x, subject to the condition that the value of the field be as small as possible at the point z, corresponding maybe to a school or a hospital in the area.


The quest for the polynomial P, and the minimal degree D of it, turn out to be related with deep results and open questions in the theory of line arrangements. Indeed, by the classical duality between points and lines in the projective plane, the set of points X gives rise to a line arrangement AX. The splitting type of a 2-vector bundle associated with the line arrangement AX plays a key role in this question, see [1] and [2] for details.
It also points to unexpected relations with Terao's Conjecture on free arrangements, as discussed in [3].

References:

[1] T. Abe, A. Dimca: On the splitting types of bundles of logarithmic vector fields along plane curves, arXiv:1706.05146

[2] D. Cook II, B. Harbourne, J. Migliore, U. Nagel: Line arrangements and configurations of points with an unusual geometry, arXiv: 1602.02300.

[3] A. Dimca: Hyperplane Arrangements: An Introduction, Universitext, Springer-Verlag, 2017, 192+viii pages.