On Waring’s problem for integral quadratic forms

03/08/2016 - 12:15pm
03/08/2016 - 1:10pm
Wai Kiu Chan (Wesleyan University)

In a 1930 paper, Louis Mordell posed the following question, that he called a new Waring’s Problem: can every positive definite integral quadratic form in n variables be written as a sum of n + 3 squares of integral linear forms? A few years later Chao Ko answered Mordell’s question in the affirmative when n ≤ 5 but provided an example of a 6-variable positive integral quadratic form which cannot be written as a sum of any squares of linear forms. In her 1992 thesis, Maria Icaza defined g_Z(n) to be the smallest integer such that if an integral quadratic form in n variables can be written as a sum of squares of integral linear forms, then it must be written as a sum of g_Z(n) squares of integral linear forms. She showed that g_Z(n) is finite and provided an upper bound which is at least exponential in n^(n^2). In this talk, we will survey some more recent results on g_Z(n) and explain our work on an upper bound on g_Z(n) which is at most exponential in n^{1/2}. Generalization to integral hermitian forms will also be discussed. This is a joint work with Constantin Beli, Maria Icaza, and Jingbo Liu.

Millikan 2099, Pomona College