# Lecture 9: Polynomial Remainder Sequences

### Date: Mar. 1

### Background Material and Further Resources

- Review Euclidean algorithm, seminorms, primitive parts and content.

### Reading

- Handout: "Computer Algebra of Polynomials"
- Chap. 7 (sec. 1 - 3)

### Topics

- Pseudo-remainder and its computing time.
- Polynomial Remainder Sequences (PRS) and similarity.
- Pseudo-remainder PRS and its computing time.
- Determinant polynomials and the pseudo-remainder.
- Determinant bound.
- Subresultants and the fundamental theorem of PRS.
- Primitive PRS and its computing time.
- Reduced PRS and Subresultant PRS.
- Exercise: A modular polynomial multiplication algorithm.

### Maple Worksheet

- prs.mws
(PRS and subresultant algorithms and examples)

### SACLIB Programs

- IPRPRS - Integral polynomial reduced polynomial remainder sequence.
- IPSPRS - Integral polynomial subresultant polynomial remainder
sequence.

Created: Feb. 29, 2000 by
jjohnson@mcs.drexel.edu