Appell Root Derivative Chain

Abstract

A strict polynomial operator budget controls the derivative chain at an actual root.

Theorem 1.1 (Conditional polynomial operator estimate).

$$\begin{array}{r}\forall q\in \mathbb{N},\forall P\in \mathbb{C}[X],\forall b\in \mathbb{C},\forall s\in \mathbb{C},\forall a\in \mathbb{R},\forall g\in (\mathbb{N}\to \mathbb{C}),\\ 1\le q\Rightarrow \mathrm{natDegree}(P)=q\Rightarrow P(b)=0\Rightarrow s\ne 0\Rightarrow 0<a\Rightarrow \\ s^2{P}^{(2)}-X{P}^{(1)}+qP=\sum _{2\le m<q}{g}_m{P}^{(m+1)}\Rightarrow \\ \frac{\Vert b\Vert }{\Vert s{\Vert }^{2}a}+\frac{q}{\Vert s{\Vert }^2{a}^2}+\sum _{2\le m<q}\frac{\Vert g_m\Vert a^{m-1}}{\Vert s{\Vert }^2}<1\Rightarrow \\ P^{(1)}(b)\ne 0\wedge (\forall m\in \mathbb{N},1\le m\Rightarrow m\le q\Rightarrow \Vert \frac{P^{(m)}(b)}{P^{(1)}(b)}\Vert \le a^{m-1})\end{array}$$

Proof. Machine-checked in Lean as D5/S3/Zeros/Jensen/AppellRootDerivativeChain.actual_root_derivative_chain (✓ std3). ∎

Source. Repository-derived.

Commentary.

P is a complex polynomial of natural degree q. The point b is an actual root, s is a nonzero complex number, a is a positive real number, and g is an arbitrary sequence of complex coefficients. P^(m) denotes the m-fold polynomial derivative, with P^(0)=P. The operator equation is equality of polynomials, not merely equality at b. Every sum ranges over the natural indices 2 <= m < q. Norms are complex absolute values, and the row inequality is real.

Differentiate the operator equation k times and evaluate at b. The product rule gives s^2 P^(k+2)(b) = b P^(k+1)(b) - (q-k) P^(k)(b) + sum g_m P^(k+m+1)(b), for k+2 <= q. Derivatives above q vanish. The top derivative is the nonzero constant q! times the leading coefficient; monicity is unnecessary.

The finite sequence |P^(j)(b)|/a^j, for 0 <= j <= q, has a positive attained maximum M. Its zeroth entry is zero. If an index k+2 attains M, the differentiated equation and triangle inequality bound that entry by B M, where B is the displayed row budget. Since B < 1, this is impossible. The maximizing index is therefore one, giving a nonzero first derivative and all displayed ratios. For q=1 the higher-index range and coefficient sum are empty.

This is the polynomial implication used in LC359–LC361 of the repository’s Romik analytic note. For its Appell specialization, P=R_(q,t)=M_t(partial_z) z^q, s^2=v chi’, and a=8 sqrt(q)/|s|. The g_m are ordinary coefficients of the logarithmic remainder; in the cumulant convention g_m=kappa_(m+1)/m!. The actual source ODE and its coefficient and root-location budgets must be established separately. This theorem supplies no global node signs, RH or Robin conclusion, and makes no claim of literature originality.

References

• Truth anchor: D5/S3/Zeros/Jensen/AppellRootDerivativeChain.actual_root_derivative_chain