Bounding the degrees of a minimal $\mu$-basis for a rational surface parametrization

TL;DR

This paper investigates degree bounds of minimal μ-bases for rational surface parametrizations, establishing polynomial degree bounds and conditions for tighter bounds, which aids in understanding the algebraic complexity of such parametrizations.

Contribution

The paper proves the existence of degree bounds for minimal μ-bases of rational surfaces and provides tighter bounds under specific conditions.

Findings

Existence of μ-bases with degree bounded by O(d^{33})

Tighter bounds are achievable under additional assumptions

Results contribute to understanding algebraic complexity of rational surface parametrizations

Abstract

In this paper, we study how the degrees of the elements in a minimal $\mu$-basis of a parametrized surface behave. For an arbitrary rational surface parametrization $P(s,t)=(a_1(s,t),a_2(s,t),a_3(s,t),a_4(s,t)) \in \mathbb{F}[s,t]^4$ over an infinite field $\mathbb{F}$, we show the existence of a $\mu$-basis with polynomials bounded in degree by $O(d^{33})$, where $d=\max(\text{deg}(a_1),\text{deg}(a_2), \text{deg}(a_3), \text{deg}(a_4))$. Under additional assumptions we can obtain tighter bounds.