Gian Cordana Sanjaya, University of À¶Ý®ÊÓƵ
Density of Special Classes of Polynomials with Squarefree Discriminant
In this talk, we discuss the problem of determining the local densities of monic integer polynomials of fixed degree with squarefree discriminant, with some conditions on the coefficients. Previously, Yamamura solved the case where no extra conditions are imposed, and Bhargava, Shankar, and Wang proved that the global density of general monic integer polynomials (of fixed degree) with squarefree discriminant is equal to the product of the local densities.
Our main case of interest is when the first two non-leading coefficients are fixed. However, we also obtain results on other cases, such as the case where the constant coefficient is fixed. Moreover, the technique used here allows us to compute the local densities exactly.
This talk is based on the recent preprint: https://arxiv.org/abs/2505.06820
MC 5417