Mathematical Sciences Colloquium

The talk will focus on the classical question of finding a "canonical form" for degree n polynomials in one variable. For example, the quadratic polynomial x² + ax + b can be transformed into one of the form x² + c by completing the square. The latter can be viewed as a canonical form which depends on only one parameter (namely, c) rather than the two we started out with (namely, a and b). In general, we will show that any canonical form for degree n polynomials depends on at least [n/2] parameters. This result (obtained in collaboration with J. Buhler) generalizes a theorem of Felix Klein on quintic polynomials and is related to Hilbert's 13-th problem.