Academic Report of the School of Mathematical Sciences [2025] No. 089

(High-level University Construction Series Report No.1111)

Lecture Title: On the number of zeros of diagonal quartic forms over finite fields

Lecturer: Professor Hong Shaofang (Sichuan University)

Lecture Time: September 30, 2025, 14:00-15:00

Venue: Huiwen Building, Room 2331

Abstract:

Let $\mathbb{F}_q$ be the finite field of $q=p^m\equiv 1\pmod 4$ elements with $p$ being an odd prime and $m$ being a positive integer. For $c, y\in\mathbb{F}_q$ with $y\in\mathbb{F}_q^*$ non-quartic, let $N_n(c)$ and $M_n(y)$ be the numbers of zeros of $x_1^4+...+x_n^4=c$ and $x_1^4+...+x_{n-1}^4+yx_n^4=0$, respectively. In 1979, Myerson used Gauss sum and exponential sum to show that the generating function $\sum_{n=1}^{\infty}N_n(0)x^n$ is a rational function in $x$ and presented its explicit expression. In this talk, we make use of the cyclotomic theory and exponential sums to show that the generating functions $\sum_{n=1}^{\infty}N_n(c)x^n$ and $\sum_{n=1}^{\infty}M_{n+1}(y)x^n$ are rational functions in $x$. We also obtain the explicit expressions of these generating functions. Our result extends Myerson's theorem gotten in 1979.

Lecturer's Biography:

Hong Shaofang is a Professor and Doctoral Supervisor at the School of Mathematics, Sichuan University. He is an Outstanding Talent of the New Century of the Ministry of Education and an Academic and Technical Leader of Sichuan Province. He is mainly engaged in undergraduate teaching and graduate training, and his research focuses on number theory, arithmetic geometry, and coding theory. He has successively presided over more than 10 vertical projects, including the National Natural Science Foundation of China and the Doctoral Program Foundation of the Ministry of Education. He has published over 100 academic papers in well-known domestic and international mathematical journals. He has visited many universities and research institutions in countries such as the United States, France, Israel, Japan, and South Korea, as well as in regions including Hong Kong and Taiwan (China), on multiple occasions. In the fields of number theory and information security, he has supervised more than 60 graduated masters and 20 graduated doctors, among whom many have been promoted to senior professional titles (professor level).

Faculty and students are welcome to attend!

Inviter: Wang Yingnan

School of Mathematical Sciences

September 22, 2025