Speaker: Robin Pemantle, Department of Mathematics, U. Penn

Title: Solving equations

Abstract: At some point, every math student discovers or is taught that solving simultaneous linear equations is completely algorithmic. Most don’t know, however, that solving simultaneous polynomial equations is equally well understood.  The solutions are algebraic numbers and algebraic functions.  These may be represented by finite data (collections of integers) and manipulated (added, multiplied, simplified).  Not only is this all in principle algorithmic but it has been implemented in computer algebra systems such as Maple, Mathematica and Sage, and is becoming very widely  used.  If there is time, I will discuss the extension of this to the (noncommutative) Weyl algebra, which allows one to compute in the much larger class of D-finite functions, which solve holonomic systems of linear differential equations with coefficients that are polynomials in the corresponding variables.