Exercise 1, § 3

a) Show that, for all i between 1 and l, there exists a unique derivation D_i of A[P] satisfying the following conditions:

a₁) D_i is A-linear.

a₂) D_i(e^{ω_j}) = δ_{ij}e^{ω_j} (δ_{ij} being the Kronecker symbol).

b) Let (x_i)_{1 ≤ i ≤ l} be a family of elements of A[P]^W satisfying the condition of Theorem 1. Show that

(Show that det(D_i(x_j)) is anti-invariant and has maximal term e^ρ, hence the result if 2 is not a zero divisor in A. Treat the general case by using the principle of permanence of algebraic identities.) ^3