Exact renewal laws for minimal common-denominator profiles in simultaneous Laurent-series approximation

Abstract

Let $\alpha_1,\ldots,\alpha_r$ be independent Haar-random fractional Laurent series over $\mathbb{F}_q$, and let $L_r(n)$ be the least coefficient length of a polynomial denominator that simultaneously cancels the first $n$ negative coefficients. We prove that the minimal kernel is a line and that the residual vectors revealed immediately after the stopping times $T_n=n+L_r(n)-1$ are iid uniform on $\mathbb{F}_q^r$. Hence the jump indicators of $L_r(n)$ are iid Bernoulli variables with parameter $1-q^{-r}$; conditionally on a jump, the residual direction is uniform on $\mathbb P^{r-1}(\mathbb{F}_q)$. We also give an exact kernel-growth clock for positive jump sizes and a geometric tail bound uniform in the depth; for two series the jump is decided at the first or second kernel-growth epoch with probabilities $q^{-1}$ and $1-q^{-1}$. The marked renewal law yields exact binomial and fluctuation laws in the depth variable and the density of newly attained minimal denominator lengths $\frac{1-q^{-r}}{r}$ in the coefficient-length variable. For $r=1$ this is the classical iid partial-quotient degree law in the depth coordinate, for which we give an exact dictionary. The new probabilistic content is the simultaneous common-denominator law for $r\ge2$. We also establish exact profile-correspondence and record-duality formulas with joint linear complexity.