Order-Degree-Height Surfaces for Linear Operators

TL;DR

This paper introduces order-degree-height surfaces for linear operators with polynomial coefficients, extending the traditional order-degree trade-off by incorporating coefficient size, and deriving relationships among these three parameters.

Contribution

It extends the classical order-degree analysis by including height, providing a new geometric perspective through order-degree-height surfaces.

Findings

Derived relationships between order, degree, and height for specific cases

Extended the order-degree curve to a surface incorporating height

Provided insights into coefficient size constraints in linear operators

Abstract

It is known for linear operators with polynomial coefficients annihilating a given D-finite function that there is a trade-off between order and degree. Raising the order may give room for lowering the degree. The relationship between order and degree is typically described by a hyperbola known as the order-degree curve. In this paper, we add the height into the picture, i.e., a measure for the size of the coefficients in the polynomial coefficients. For certain situations, we derive relationships between order, degree, and height that can be viewed as order-degree-height surfaces.