# When does the constant term in the following expansion is nonzero?

Dyson's Theorem

The constant term in the expansion of $$\prod_{1\leq i\neq j\leq n}\left(1-\frac{x_i}{x_j}\right)^{a_i}$$ is the multinomial coefficient $$\frac{(a_1+\cdots+a_n)!}{a_1!\cdots a_n!},$$ where $a_1,\cdots,a_n$ are nonnegative integers.

My question is for general $$F(x_1,\cdots,x_n)=\prod_{1\leq i\neq j\leq n}\left(1-\frac{x_i}{x_j}\right)^{a_{ij}}$$ where $a_{ij}$ are nonnegative integers, are there any results or theories about when the constant term in the expansion of $F(x_1,\cdots,x_n)$ is nonzero?

• There are numerous papers on constant terms of such products, but of course no general full answer. – Fedor Petrov Mar 24 '17 at 11:40
• Hello, Fedor Petrov, would you please tell me how to search the papers on constant terms of such products? – user173856 Mar 24 '17 at 11:50
• In this paper of G. Károlyi, Z. Nagy, V. Volkov and myself a general technology is presented, it also contains a survey and bibliography sciencedirect.com/science/article/pii/S0001870815000778 – Fedor Petrov Mar 24 '17 at 11:54
• Fedor Petrov, thank you for your help! – user173856 Mar 26 '17 at 5:34