5.2 Why the constant term varies

The constant term of \(P_N(x; \theta )\) is \((-1)^N e_N = (-1)^N \prod _{k=0}^{N-1} r_k(\theta )\). While the product of roots can vary with \(\theta \), Newton’s identities only constrain \(e_N\) using \(p_1, \ldots , p_N\). Since \(p_N\) (the sum of \(N\)-th powers) may depend on \(\theta \) when \(j = N \geq N\), there is no contradiction. Indeed, explicit calculation shows the constant term does vary.