# On Some Problems on Polynomial Interpolation in Several Variables

2020

Dissertation

Ph.D.

## Degree Name

Doctor of Philosophy (Ph.D.)

## Degree Granting Department

Mathematics and Statistics

## Major Professor

Boris Shekhtman, Ph.D.

## Committee Member

Xiang-dong Hou, Ph.D.

## Committee Member

Sherwin Kouchekian, Ph.D.

## Committee Member

Garrett Matthews, Ph.D.

## Committee Member

Razvan Teodorescu, Ph.D.

## Keywords

Hermite, Projection, Smoothable, Varieties, Projector, Polynomial Approximation

## Abstract

Polynomial approximation is a long studied process, with a history dating back to the 1700s, At which time Lagrange, Newton and Taylor developed their famed approximation methods. At that time, it was discovered that every Taylor projection (projector) is the pointwise limit of Lagrange projections. This leaves open a rather large and intriguing question, What happens in several variables?

To this end we define a linear idempotent operator to be an ideal projector whenever its kernel is an ideal. No matter the number of variables, Taylor projections and Lagrange projections are always ideal projectors, and it is well known that in one variable, that, not only Taylor projections, but every ideal projector, is the pointwise limit of Lagrange projections. This is also true in two variables, but false in three or more variables. We call the projectors which are the pointwise limits of Lagrange projectors, Hermite projectors. As it turns out, the Hermite projectors happen to be exactly those projectors whose kernels are something algebraic geometers refer to as smoothable. The question of which ideals are smoothable is also an open question in algebraic geometry. This correlation, provides the humble researcher with a whole new slew of tools to apply to problems.

It is the aim of this dissertation to provide a field map to this interesting environment, in which some problems, previously intractable, can be approached with renewed vigor. One such problem, unstudied except for some very specific cases, is, given sets V1,...,Vn, is it possible to find a polynomial p which interpolates each polynomial pi on Vi. We present the results of a paper which was submitted for publication providing a generalized extension of a theorem by W.K. Hayman and Z. G. Shandze. For the second part of the dissertation we present a result of a second paper that was submitted for publication, in which we make a useful contribution to a question of Carl de Boor, which ideal projectors are Hermite.

In the first part of this dissertation we find that the answer to the question about interpolation on sets is: sometimes. We will provide some conditions under which it is, and is not, possible to do this. One of these conditions is the aforementioned extension of a result of W.K. Hayman and Z. G. Shandze. In the secondpart we make a contribution to the question of which projectors are Hermite. The Laskar-Noether theorems hows that every ideal has a unique minimal decomposition into primary ideals, we prove that if J1,...,Jk is the minimal primary decomposition of J, and P, P1,...,Pk are ideal projectors with kernels J, J1,...,Jk respectively, then P is Hermite if and only if each Pi is Hermite. In the language of algebraic geometry, an equivalent statement, is that the ideal J is smoothable, if and only if each Ji is smoothable.

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