Abstract.  

The analogue of the method of long division in the ring of integers is the division
algorithm for the ring of polynomials k[x] (in a single variable x, with coefficients in a
field k). This division algorithm can be used to show that each ideal in k[x] is generated
by a single polynomial, that is, k[x] is a principal ideal domain. For rings k[x_1,..., x_n]
of polynomials in more than one variable, it is easy to show that they are not principal
ideal domains. What can one say about ideals in k[x_1,..., x_n]? In this second of a series of two talks, we will answer this question by proving Hilbert’s Basis Theorem (HBT) for polynomial rings k[x_1,..., x_n] as a consequence of a good multivariable division algorithm. Concomitantly we will arrive at the existence of Groebner bases, sets of ideal generators which act particularly nicely with respect to multivariable long division.  Along the way, we will see how Groebner bases rectify some surprising features of more naïve attempts to implement polynomial long division in many variables. Finally, we will give some powerful ring-theoretic and geometric consequences of HBT and Groebner bases.