Banded quadratic digit functions along irreducible polynomials over finite fields

Let $q$ be an odd prime power and let ${\text{F}}_q$ be the finite field with $q$ elements. Let $\mathcal{P}(n)$ be the set of monic irreducible polynomials of degree $n$ over ${\mathbb{F}}_q$. For $f=t^n+f_{n-1}{t}^{n-1}+\cdots +f_0\in \mathcal{P}(n)$, fix coefficients $c_0,\dots ,c_m\in {\mathbb{F}}_q$ with $c_m\mathrm{\ne }0$ and put $$Q_A(f)=\sum _{j=0}^m{c}_j\sum _{i=j}^n{f}_i{f}_{i-j}+{\mathrm{\ell }}_n(f),$$ where ${\mathrm{\ell }}_n$ is an arbitrary linear form in the coefficients of $f$ and $f_n=1$. We prove that $Q_A$ is equidistributed on $\mathcal{P}(n)$: for every $\gamma \in {\mathbb{F}}_q$, $$\mathrm{\#}\{f\in \mathcal{P}(n):Q_A(f)=\gamma \}=\frac{\mathrm{\#}\mathcal{P}(n)}{q}+O_A(q^{19n\mathrm{/}20+o(n)}),$$ as $n\to \mathrm{\infty }$, with $q$ and the quadratic band fixed. This extends the finite-field Rudin–Shapiro result from nearest-neighbour correlations to arbitrary fixed symmetric Laurent symbols. The proof combines Vaughan’s identity with rank estimates for Toeplitz forms; the main new ingredient is an averaged rank-defect estimate for reciprocal symbols in the central Type I range.