The sum of four squares (SFS), which was proved by Lagrange in 1770,
states that every natural number can be written as a sum of four squares.
The theme of this talk is various generalizations of this beautiful result.
There are two classical generalizations of Lagrange's SFS.
The one is Cauchy's polygonal number theorem which can be viewed
as a horizontal generalization of the SFS, and the other is
Waring's problem which can be viewed as a higher dimensional
generalization of the SFS. As further generalizations,
we introduce two types of Waring's problem, namely Waring's problem
for integer valued polynomials (an arithmetic generalization)
and Waring's problem for polytope numbers (a geometric generalization).
We next introduce the circle method which was initiated
by Hardy and Littlewood in 1920's.
We will use the circle method to illustrate that Waring's problem
for polytope numbers is more natural than that for integer valued polynomials.
〒840-8502 佐賀市本庄町1 佐賀大学理工学部数理科学科
TEL 0952-28-8521 FAX 0952-28-8501
校時 金 I, II, III, IV
土 I, II, III, IV
月 I, II, III, IV
火 I, II, III
I校時 金 賢光（Kim Hyun kwang）
II校時 上原 健
III校時 中原 徹