[Paper Review] Uniform estimates for smooth polynomials over finite fields

Ofir Gorodetsky|arXiv (Cornell University)|Mar 9, 2022

Analytic Number Theory Research4 citations

TL;DR

This paper establishes uniform asymptotic estimates for the number of $m$-smooth monic polynomials of degree $n$ over finite fields $\mathbb{F}_q$, showing that their smoothness probability asymptotically matches that of random permutations on $n$ elements when $m \geq (2+\varepsilon)\log_q n$, uniformly as $q^n \to \infty$. The key result resolves a polynomial analogue of a conjecture by Hildebrand, proving the range $m \geq (2+\varepsilon)\log_q n$ is sharp and providing sharp error terms via saddle-point analysis and comparison to the Dickman function.

ABSTRACT

We establish new estimates for the number of $m$-smooth polynomials of degree $n$ over a finite field $\mathbb{F}_q$, where the main term involves the number of $m$-smooth permutations on $n$ elements. Our estimates imply that the probability that a random polynomial of degree $n$ is $m$-smooth is asymptotic to the probability that a random permutation on $n$ elements is $m$-smooth, uniformly for $m\ge (2+\varepsilon)\log_q n$ as $q^n o \infty$. This should be viewed as an unconditional analogue of works of Hildebrand and of Saias in the integer setting, which assume the Riemann Hypothesis. Moreover, we show that the range $m \ge (2+\varepsilon)\log_q n$ is sharp; this should be viewed as a resolution of a (polynomial analogue of a) conjecture of Hildebrand. As an application of our estimates, we determine the rate of decay in the asymptotic formula for the expected degree of the largest prime factor of a random polynomial.

Motivation & Objective

• To establish uniform asymptotic estimates for the number of $m$-smooth monic polynomials of degree $n$ over $\mathbb{F}_q$.
• To compare the smoothness probability of random polynomials in $\mathbb{F}_q[T]$ to that of random permutations in $S_n$, showing asymptotic equivalence under certain conditions.
• To resolve a polynomial analogue of a conjecture by Hildebrand on the sharpness of the threshold $m \geq (2+\varepsilon)\log_q n$ for smoothness probabilities.
• To determine the rate of decay in the expected degree of the largest prime factor of a random polynomial using the derived estimates.

Proposed method

• The paper uses saddle-point analysis to estimate the generating function for $m$-smooth polynomials over $\mathbb{F}_q$, comparing it to the generating function for $m$-smooth permutations.
• It introduces a parameter $x = x_{n,m} > 0$ satisfying $\sum_{j=1}^m x^j = n$, which controls the asymptotic behavior of the smoothness probability.
• The main term in the asymptotic estimate involves the Dickman function $\rho(u)$, with $u = n/m$, and the error terms are controlled via integral estimates and bounds on the difference between discrete and continuous averages.
• A key component is the function $G_q(x)$, defined as a product over irreducible polynomials of degree $\leq m$, which captures the deviation from the permutation model.
• The proof relies on a recursive relation for the smoothness probability in $S_n$, expressed via $\Delta(n,m)$, and establishes positivity and lower bounds for $\Delta(n,m)$ using integral representations and asymptotic expansions of $\rho(u)$.
• The analysis includes careful estimation of the error terms in the ratio $\mathbb{P}(f_n \text{ is } m\text{-smooth}) / \mathbb{P}(\pi_n \text{ is } m\text{-smooth})$, showing uniform convergence to 1 as $q^n \to \infty$.

Experimental results

Research questions

• RQ1 Does the probability that a random monic polynomial of degree $n$ over $\mathbb{F}_q$ is $m$-smooth asymptotically match the probability that a random permutation on $n$ elements is $m$-smooth, uniformly in $n$ and $q$, for $m \geq (2+\varepsilon)\log_q n$?
• RQ2 Is the threshold $m \geq (2+\varepsilon)\log_q n$ sharp for the asymptotic equivalence of polynomial and permutation smoothness probabilities?
• RQ3 What is the rate of decay of the expected degree of the largest irreducible factor of a random polynomial over $\mathbb{F}_q$?
• RQ4 How do the error terms in the asymptotic estimate for $\mathbb{P}(f_n \text{ is } m\text{-smooth})$ compare to those in the permutation model, and can they be improved uniformly?
• RQ5 Can the comparison between polynomial and permutation smoothness be extended to the regime $m \sim (1+\varepsilon)\log_q n$, and what modifications to the main term are needed?

Key findings

• For $m \geq (2+\varepsilon)\log_q n$, the probability that a random polynomial of degree $n$ over $\mathbb{F}_q$ is $m$-smooth is asymptotically equivalent to the probability that a random permutation on $n$ elements is $m$-smooth, uniformly as $q^n \to \infty$, with relative error $\ll_{\varepsilon} 1/(nq)^{c_\varepsilon}$.
• The range $m \geq (2+\varepsilon)\log_q n$ is sharp: for $m < (2+\varepsilon)\log_q n$, the asymptotic equivalence fails, confirming a conjecture of Hildebrand in the function field setting.
• The error term in the ratio $\mathbb{P}(f_n \text{ is } m\text{-smooth}) / \mathbb{P}(\pi_n \text{ is } m\text{-smooth})$ is bounded by $O_{\varepsilon}\left(\frac{un^{(1+a)/m}}{q^{\lceil (m+1)/2 \rceil}}\right)$ with $a = \mathbf{1}_{2\mid m}$, improving upon prior error bounds.
• For $m \geq 6\log n$, the error term is $O\left(\frac{u\log(u+1)}{mq^{\lceil (m+1)/2 \rceil}}\right)$, which is significantly smaller than previous bounds of order $u\log(u+1)/m$ or $1/u$.
• When $m \sim (1+\varepsilon)\log_q n$, the main term in the asymptotic ratio is modified by a factor $G_q(x)$, which accounts for the discrepancy between the polynomial and permutation models.
• The paper establishes a lower bound $\Delta(n,m) \gg u\log u / m$ for $m \leq n/2$, where $\Delta(n,m)$ measures the relative difference in smoothness probabilities between permutations and polynomials, confirming the positivity and growth of this difference in the relevant regime.