Degree-optimal moving frames for rational curves

Abstract

A $\mathit{\text{moving frame}}$ at a rational curve is a basis of vectors moving along the curve. When the rational curve is given parametrically by a row vector $\mathbf{a}$ of univariate polynomials, a moving frame with important algebraic properties can be defined by the columns of an invertible polynomial matrix $P$, such that $\mathbf{a} P=[\gcd(\mathbf{a}),0\ldots,0]$. A $\mathit{\text{degree-optimal moving frame}}$ has column-wise minimal degree, where the degree of a column is defined to be the maximum of the degrees of its components. Algebraic moving frames are closely related to the univariate versions of the celebrated Quillen-Suslin problem, effective Nullstellensatz problem, and syzygy module problem. However, this paper appears to be the first devoted to finding an efficient algorithm for constructing a degree-optimal moving frame, a property desirable in various applications. We compare our algorithm with other possible approaches, based on already available algorithms, and show that it is more efficient. We also establish several new theoretical results concerning the degrees of an optimal moving frame and its components. In addition, we show that any deterministic algorithm for computing a degree-optimal algebraic moving frame can be augmented so that it assigns a degree-optimal moving frame in a $GL_n(\mathbb{K})$-equivariant manner. This crucial property of classical geometric moving frames, in combination with the algebraic properties, can be exploited in various problems.