#### Conference abstracts

Session A2 - Computational Algebraic Geometry

July 11, 17:30 ~ 17:55 - Room B5

## Degree-Optimal Moving Frames for Rational Curves

### Irina Kogan

### North Carolina State University, USA - iakogan@ncsu.edu

We present an algorithm that, for a given vector $\mathbf a$ of $n$ relatively prime polynomials in one variable over an arbitrary field, outputs an $n\times n$ invertible matrix $P$ with polynomial entries, such that it forms a degree-optimal moving frame for the rational curve defined by $\mathbf a$. The first column of the matrix $P$ consists of a minimal-degree Bézout vector (a minimal-degree solution to the univariate effective Nullstellensatz problem) of $\mathbf a$, and the last $n-1$ columns comprise an optimal-degree basis, called a $\mu$-basis, of the syzygy module of $\mathbf a$. To develop the algorithm, we prove several new theoretical results on the relationship between optimal moving frames, minimal-degree Bézout vectors, and $\mu$-bases. In particular, we show how the degree bounds of these objects are related. Comparison with other algorithms for computing moving frames and Bézout vectors will be given, however, we are currently not aware of another algorithm that produces an optimal degree moving frame or a Bézout vector of minimal degree.

Joint work with Hoon Hong (North Carolina State University, USA), Zachary Hough (North Carolina State University, USA) and Zijia Li (Joanneum Research, Austria).