Behaviour of the least root modulo a prime of a polynomial

TL;DR

This paper proposes a conjecture about the distribution of the smallest root modulo primes for polynomials with roots that are algebraically independent, focusing on primes where the polynomial fully splits.

Contribution

It introduces a new conjecture on the distribution of the least root modulo primes for certain polynomials, expanding understanding of polynomial root behavior over finite fields.

Findings

Conjecture on the distribution of the least root modulo primes.

Analysis of polynomials without linear relations among roots.

Focus on primes where the polynomial fully splits.

Abstract

For a polynomial $f(x)\in\mathbb Z[x]$ without non-trivial linear relations among roots, we propose a conjecture on the distribution of the least root $r_p$ ($r_p\in\mathbb Z,\,0\le r_p<p)$ of $f(x)\equiv0\bmod p$ where $p$ runs over the set of primes such that $f(x)$ modulo $p$ is fully splitting.

Equations

$$f(x)=x^n+a_{n-1}{x}^{n-1}+\cdots +a_0$$

$$Sp{l}_X(f):=\left\{p\le X\mid f(x)\text{ is fully splitting modulo }p\right\}$$

$$f(x)\equiv \prod _{i=1}^n(x-r_i)\mathrm{mod}p,$$

$$0\le r_1\le r_2\le \cdots \le r_n<p.$$

$$\sum _{i=1}^n{m}_i{\alpha }_i=m(m_i,m\in \mathbb{Q}).$$

$${\hat{\mathfrak{D}}}_n:=\{(x_1\dots ,x_n)\in [0,1{)}^n\mid 0\le x_1\le \cdots \le x_n<1,\sum _{i=1}^n{x}_i\in \mathbb{Z}\}$$

$$P{r}_D(f,X):=\frac{\#\{p\in Sp{l}_X(f)\mid (r_1/p,\dots ,r_n/p)\in D\}}{\#Sp{l}_X(f)},$$

$$P{r}_D(f):=\underset{X\to \infty }{\lim }P{r}_D(f,X)=\frac{vol(D\cap {\hat{\mathfrak{D}}}_n)}{vol({\hat{\mathfrak{D}}}_n)}.$$

$$D^a:=\{(x_1,\dots ,x_n)\in [0,1{)}^n\mid a\le x_1\}(0\le a<1),$$

$$D_a:=\{(x_1,\dots ,x_n)\in [0,1{)}^n\mid x_1\le a\}(0\le a<1).$$

$$vol(D_a\cap {\hat{\mathfrak{D}}}_n)+vol(D^a\cap {\hat{\mathfrak{D}}}_n)=vol({\hat{\mathfrak{D}}}_n).$$

$$\frac{1}{(n-1)!}\sum _{\genfrac{}{}{0ex}{}{2\le i\le n}{\max (ia,i-n+1)\le k\le i-1}}(-1{)}^{n+i}(\genfrac{}{}{0ex}{}{n}{i})(k-ia{)}^{n-1}.$$

$$\{\begin{array}{ll}2a& (0\le a\le 1/2),\\ 1& (1/2\le a\le 1).\end{array}$$

$$\{\begin{array}{ll}3a-3a^2& (0\le a\le 1/3),\\ (1+3a^2)/2& (1/3\le a\le 1/2),\\ -1+6a-9a^2/2& (1/2\le a\le 2/3),\\ 1& (2/3\le a\le 1).\end{array}$$

$$\{\begin{array}{ll}4a^3-6a^2+4a& (0\le a\le 1/4),\\ -20a^3/3+2a^2+2a+1/6& (1/4<a\le 1/3),\\ 34a^3/3-16a^2+8a-1/2& (1/3<a\le 1/2),\\ -22a^3/3+12a^2-6a+11/6& (1/2<a\le 2/3),\\ 32a^3/3-24a^2+18a-7/2& (2/3<a\le 3/4),\\ 1& (3/4<a<1).\end{array}$$

$$\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|}\hline\cr&n=2&3&4&5&6&7&8&9&10\\ \hline\cr m=7&1(3)&1(3)&2(3)&9(3)&1(2)&5(2)&2(1)&2(1)&8(1)\\ \hline\cr 8&4(4)&6(4)&1(3)&6(3)&1(2)&2(2)&3(2)&1(1)&3(1)\\ \hline\cr 9&4(5)&3(4)&3(4)&1(3)&1(3)&8(3)&1(2)&3(2)&1(1)\\ \hline\cr\end{array}$$

$$U_n(x):=\frac{1}{n!}\sum _{i=0}^n(-1{)}^i(\genfrac{}{}{0ex}{}{n}{i})\max (x-i,0{)}^n.$$

$$vol(D^a\cap {\hat{\mathfrak{D}}}_n)$$

$$=vol(\{(x_1,\dots ,x_n)\mid a\le x_1\le \cdots <x_n<1,\sum x_i\in \mathbb{Z}\})$$

$$=\frac{1}{n!}vol(\{(x_1,\dots ,x_n)\mid a\le x_1,\dots ,x_n<1,\sum x_i\in \mathbb{Z}\})$$

$$\displaystyle=\frac{1}{n!\cos\theta}\sum_{k=1}^{n-1}vol\left(\left\{(x_{1},\dots,x_{n-1})\left|\begin{array}[]{l}a\leq x_{1},\dots,x_{n-1}<1,\\ k-1<\sum_{i=1}^{n-1}x_{i}\leq k-a\end{array}\right.\right\}\right).$$

$${\int }_{\genfrac{}{}{0ex}{}{a\le x_1,\dots ,x_{n-1}<1,}{k-1<{\sum }_{i=1}^{n-1}{x}_i\le k-a}}d{x}_1\dots d{x}_{n-1}$$

$$=$$

$$(1-a{)}^{n-1}\sum _{k=1}^{n-1}\left\{U_{n-1}(\frac{k-na}{1-a})-U_{n-1}(\frac{k-na}{1-a}-1)\right\}$$

$$=$$

$$-\sum _{\genfrac{}{}{0ex}{}{i=0}{i<\frac{k-na}{1-a}-1}}^{n-1}(-1{)}^i(\genfrac{}{}{0ex}{}{n-1}{i}\left){\left(\frac{k-na}{1-a}-1-i\right)}^{n-1}\right\}$$

$$=$$

$$+\sum _{\genfrac{}{}{0ex}{}{j=1}{j<\frac{k-na}{1-a}}}^n(-1{)}^j(\genfrac{}{}{0ex}{}{n-1}{j-1}\left){\left(\frac{k-na}{1-a}-j\right)}^{n-1}\right\}$$

$$=$$

$$+\sum _{k=1}^{n-1}\sum _{\genfrac{}{}{0ex}{}{j=1}{j<\frac{k-na}{1-a}}}^{n-1}(-1{)}^j(\genfrac{}{}{0ex}{}{n-1}{j-1}\left){\left(\frac{k-na}{1-a}-j\right)}^{n-1}\right\},$$

$$=$$

$$E_a:=\{(x_1,\dots ,x_n)\in [0,1{)}^n\mid x_n\le a\}.$$

$$\frac{1}{(n-1)!}\sum _{\genfrac{}{}{0ex}{}{0\le i\le n}{\max (ia,1)\le k\le n-1}}(-1{)}^i(\genfrac{}{}{0ex}{}{n}{i})(k-ia{)}^{n-1}=V^{(1-a)}.$$

$$\sum _{i=0}^n(-1{)}^{i}P(i)(\genfrac{}{}{0ex}{}{n}{i})=c_n(-1{)}^{n}n!$$

Full text

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This means that the conjectural value of the density of primes $p$ satisfying that every root $r\,(0\leq r<p)$ of $f(x)\equiv 0\bmod p$ is greater than $ap$ is equal to $V^{a}$, if a polynomial $f$ has only a trivial linear relation among roots.

Let us give more explicitly an expected density $V_{a}:=1-V^{a}$ for a domain $D_{a}$ in the case of $n=2,3,4$.

In case of $n=2$:

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In case of $n=3$:

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In case of $n=4$:

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Let us give numerical data of differences between the expected density $V^{a}$ and $Pr_{D^{a}}(f,X)$ $(a=1/(10n),\dots,(10n-1)/(10n))$ for $f=x^{n}+3x+1$, whose Galois group is isometric to the symmetric group. The following is a table111 Data were made by pari/gp. The PARI Group, PARI/GP version 2.8.0, Bordeaux, 2014, http://pari.math.u-bordeaux.fr/. of $\max_{1\leq k\leq 10n-1}|Pr_{D^{k/(10n)}}(f,X_{m})-V^{k/(10n)}|$, where an integer $X_{m}$ denotes the least prime number exceeding $10^{m}$ in $Spl(f)$.

[TABLE]

In the table, $1(3)$ for $m=7,n=2$ means $1.\cdots\times 10^{-3}=0.001\cdots$ and so on.

Before a calculation of the volume of $vol({D^{a}}\cap{\hat{\mathfrak{D}}_{n}})$, we refer to the following fundamental lemma ([1]).

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Denoting by $\theta$ the angle of two hyperplanes of $\mathbb{R}^{n}$ defined by $x_{1}+\dots+x_{n}=0$ and $x_{n}=0$, we have

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Putting $x_{i}=a+t_{i}(1-a)$, we see that the summand is

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Hence, by noting $vol(\hat{\mathfrak{D}}_{n})=1/(n!\cos\theta)$ ([2]), $V^{a}:=vol({D^{a}}\cap{\hat{\mathfrak{D}}_{n}})/vol(\hat{\mathfrak{D}}_{n})$ is equal to

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Hence we have completed the proof. $\Box$

We note that $\left.\frac{dV^{a}}{da}\right|_{a=0}=-n$.

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Then the right-hand side of (6) for $E_{a}$ is equal to

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A proof is similar to the previous theorem, using the fact that

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for every polynomial $P(x)=c_{n}x^{n}+c_{n-1}x^{n-1}+\dots+c_{0}$ of degree at most $n$.

What is the volume $vol(D\cap\hat{\mathfrak{D}}_{n})$ for

[TABLE]