Cameron L. Stewart: On integers represented by binary forms (NTWS 043)
Channel: Number Theory Web Seminar
Date: 2023-06-04

Abstract: We shall discuss the following results which are joint work with Stanley Xiao.
Let F(x,y) be a binary form with integer coefficients, degree d (greater than 2) and non-zero discriminant. There is a positive number C(F) such that the number of integers of absolute value at most Z which are represented by F is asymptotic to C(F)Z^(2/d).
Let k be an integer greater than 1 and suppose that there is no prime p such that p^k divides F(a,b) for all pairs of integers (a,b). Then, provided that k exceeds 7d/18 or (k,d) is (2,6) or (3,8), there is a positive number C(F,k) such that the number of k-free integers of absolute value at most Z which are represented by F is asymptotic to C(F,k)Z^(2/d).

Link to slides: https://drive.google.com/file/d/1xW6qB8m5IWXovLy0r5Lemv_l5_aqRLj1

Number Theory Web Seminar: https://www.ntwebseminar.org

Original air date: 
Thursday, October 15, 2020 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm China Standard Time)
Friday, October 16, 2020 (2am AEDT, 4am NZDT)