Hall 123

Newton's symmetric polynomials and some open problems

Neeraj Kumar

PDF, IMSc

The talk will be focused on power sum, elementary symmetric and complete symmetric polynomials (from Macdonald's book). Towards the end, we will
also talk about Schur polynomials.

Let "S" be a polynomial ring in several variable and "I" an ideal of "S" generated by symmetric polynomials. Then the theme under discussion will
be: irreducibly of symmetric polynomials, dimension of quotient ring S/I, in particular dimension zero ring (i.e. complete intersection rings), and primness of ideal "I". There are some very interesting open conjectures
related to complete intersection rings by Conca, Krattenthaler, Watanabe
and by Fr\"oberg and Shapiro.

We will see the motivation behind these conjectures and partial evidence in support of it, but not the complete proofs. This talk is aimed at
general audience, however occasionally technical terms and arguments may come-up from commutative algebra or from algebraic geometry, or from
number theory.