### The density of binary forms with coefficients in Zp that have roots over Qp

Linfoot Number Theory Seminar

14th October 2020, 11:00 am – 12:00 pm

Virtual Seminar, https://bristol-ac-uk-dev.zoom.us/j/97462603104

Let f be a random binary form in Zp[x,y] of a fixed degree d. We determine the density of such f that have a linear factor over Zp, i.e., a zero in P^1(Qp). We also determine the density of monic polynomials f in Zp[x] of degree d, which have a root in Zp. We show that these densities are rational functions in p, which only depend on d. We give the asymptotic result when p tends to infinity, and discuss the symmetry phenomenon that occurs. This is joint work Manjul Bhargava, John Cremona, and Tom Fisher.

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