Vieta-like identities for rational functions and Hurwitz Frobenius manifolds

Alessandro Proserpio (University of Glasgow)

Tuesday 18th March 16:00-17:00 Maths 311B

Abstract

Viéta identities famously relate the zeros of a polynomial to its coefficients.  Our interest in these relations lies in the fact that the space of such functions is naturally a Frobenius manifold. Here, polynomials will play a role in the Saito construction, and Viéta identities in the Saito flat coordinates. We show how this idea can be generalised to spaces of rational functions, and that the corresponding Frobenius manifolds are controllable from polynomial ones. The additional information is encoded into a diagonal action of the symmetric group, whose invariant ring is generated by Weyl's polarised power sums. If time permits, we will see how this applies to Dubrovin-Zhang Frobenius manifolds, and correspondingly to exponential polarised power sums. This is joint with Ian Strachan.

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