# Introduction to Integral Discriminants - Mathematical Physics

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Abstract: The simplest partition function, associated with homogeneous symmetric formsS of degree r in n variables, is integral discriminant J {n|r}S = \inte^{-Sx 1

. x n} dx 1

. dx n. Actually, S-dependence remains the same ife^{-S} in the integrand is substituted by arbitrary function fS, i.e.integral discriminant is a characteristic of the form S itself, and not of theaveraging procedure. The aim of the present paper is to calculate J {n|r} in anumber of non-Gaussian cases. Using Ward identities - linear differentialequations, satisfied by integral discriminants - we calculate J {2|3},J {2|4}, J {2|5} and J {3|3}. In all these examples, integral discriminantappears to be a generalized hypergeometric function. It depends on severalSLn invariants of S, with essential singularities controlled by the ordinaryalgebraic discriminant of S.

Author: ** A.Morozov, Sh.Shakirov**

Source: https://arxiv.org/