(Joint work with Andy O’Desky) There is a very classical formula counting the number of irreducible polynomials in one variable over a finite field. We study the analogous question in many variables and generalize Gauss’ formula. Our techniques can be used to answer many other questions about the space of irreducible polynomials in many variables such as it’s euler characteristic or euler hodge-deligne polynomial. To prove these results, we define a generalization of the classical ring of symmetric functions and use natural basis in it to help us compute the answer to the above questions.

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Last updated: 13 Sep 2024