Enumerating number fields

TL;DR

This paper develops small models of number fields to improve bounds on their quantity given degree and discriminant constraints.

Contribution

It introduces a novel approach to model construction that refines existing bounds on the number of number fields.

Findings

Derived tighter bounds for number fields with fixed degree and discriminant

Constructed explicit small models of number fields

Enhanced understanding of the distribution of number fields

Abstract

We construct small models of number fields and deduce a better bound for the number of number fields of given degree and bounded discriminant.

Equations

$$E_1=E_2=\cdots =E_r=0\text{ and }\det {\left(\partial E_i/\partial x_j\right)}_{1⩽i,j⩽r}\ne 0$$

$${\mathbf{K}}_{\mathbf{R}}=\mathbf{K}{\otimes }_{\mathbf{Q}}\mathbf{R}={\mathbf{R}}^r\times {\mathbf{C}}^s$$

$$<x,y>=\sum _{1⩽i⩽r}{x}_i{y}_i+\sum _{1⩽j⩽s}{x}_j{\bar{y}}_j+{\bar{x}}_j{y}_j.$$

$$<x,x>=\sum _{1⩽i⩽r}{x}_i^2+2\sum _{1⩽j⩽s}|x_j{|}^2.$$

$$2^r(2\pi {)}^s⩾2^n.$$

$$||x||⩾1.$$

$$v_{\mathbf{O}}=\sqrt{\det (B)}=|\det (A)|.$$

$$d_{\mathbf{K}}=\det (B)=|\det (A){|}^2=v_{\mathbf{O}}^2.$$

$$\prod _{1⩽i⩽n}||{\omega }_i||⩽v_{\mathbf{O}}=d_{\mathbf{K}}^{1/2}.$$

$$||{\omega }_i||⩽v_{\mathbf{O}}^{1/(n+1-i)}$$

$$||{\omega }_i||⩽d_{\mathbf{K}}^{1/n}$$

$$({\omega }_i{\omega }_j{)}_{1⩽i,j⩽m}$$

$$⩽d_{\mathbf{K}}^{2/n}.$$

$${\mathbf{K}}_{\mathbf{C}}=\mathbf{K}{\otimes }_{\mathbf{Q}}\mathbf{C}={\mathbf{C}}^n.$$

$$n(r+1)⩽\left(\genfrac{}{}{0ex}{}{d+r}{d}\right).$$

$${\mathbf{A}}_{\mathbf{C}}^r=\mathrm{Spec}\mathbf{C}[x_1,\dots ,x_r]\subset \mathrm{Proj}\mathbf{C}[x_0,x_1,\dots ,x_r]={\mathbf{P}}_{\mathbf{C}}^r.$$

$${\mathbf{C}}^r={\mathbf{A}}^r(\mathbf{C}).$$

$$H^0({\mathcal{O}}_{{\mathbf{P}}^r}(d))\to H^0({\mathcal{O}}_{2P}(d))$$

$${\mathcal{M}}_P^1=[(m(P_{\tau }){)}_{\tau ,m\in M},(\partial m/\partial x_1(P_{\tau }){)}_{\tau ,m\in M},(\partial m/\partial x_2(P_{\tau }){)}_{\tau ,m\in M},\dots ,(\partial m/\partial x_r(P_{\tau }){)}_{\tau ,m\in M}]$$

$${\kappa }_j=\sum _{1⩽i⩽n}{u}_{i,j}{\alpha }_i$$

$${ϵ}_{\mathbf{Q}}:\mathbf{Q}[x_1,\dots ,x_r]\to \mathbf{K}$$

$$e_{\mathbf{Q}}:\mathrm{Spec}\mathbf{K}\to {\mathbf{A}}_{\mathbf{Q}}^r\subset {\mathbf{P}}_{\mathbf{Q}}^r$$

$${ϵ}_{\mathbf{R}}:\mathbf{R}[x_1,\dots ,x_r]\to {\mathbf{K}}_{\mathbf{R}}$$

$$e_{\mathbf{R}}:\mathrm{Spec}{\mathbf{K}}_{\mathbf{R}}\to {\mathbf{A}}_{\mathbf{R}}^r\subset {\mathbf{P}}_{\mathbf{R}}^r$$

$${ϵ}_{\mathbf{C}}:\mathbf{C}[x_1,\dots ,x_r]\to {\mathbf{K}}_{\mathbf{C}}$$

$$e_{\mathbf{C}}:\mathrm{Spec}{\mathbf{K}}_{\mathbf{C}}\to {\mathbf{A}}_{\mathbf{C}}^r\subset {\mathbf{P}}_{\mathbf{C}}^r$$

$$(x_j(P_{\tau }){)}_{\tau }={\left(\sum _{1⩽i⩽n}{u}_{i,j}\tau ({\alpha }_i)\right)}_{\tau },$$

$$P_{\tau }=(\sum _{1⩽i⩽n}{u}_{i,j}\tau ({\alpha }_i){)}_{1⩽j⩽r}\in {\mathbf{C}}^r={\mathbf{A}}^r(\mathbf{C})\subset {\mathbf{P}}^r(\mathbf{C}).$$

$$[0,dn(r+1)]$$

$$V_{\mathbf{Z}}=\mathbf{Z}[x_1,\dots ,x_r{]}_d.$$

Full text

1. 1 Introduction
2. 2 Short integers
3. 3 Small models
4. 4 Proof of main results

1. Introduction

We prove the two theorems below.

Theorem 1** (Number fields have small models).**

There exists a positive constant ${\mathcal{O}}$ such that the following is true. Let ${\mathbf{K}}$ be a number field of degree $n\geqslant{\mathcal{O}}$ and discriminant $d_{\mathbf{K}}$ over ${\mathbf{Q}}$. There exist integers $r\leqslant{\mathcal{O}}\log n$ and $d\leqslant{\mathcal{O}}\log n$ such that ${d+r\choose r}\leqslant{\mathcal{O}}n\log n$ and there exists $r$ polynomials $E_{1}$, $E_{2}$, …, $E_{r}$ of degree $\leqslant d$ in ${\mathbf{Z}}[x_{1},\ldots,x_{r}]$ all having coefficients bounded in absolute value by $n^{{\mathcal{O}}\log n}d_{\mathbf{K}}^{{\mathcal{O}}\frac{\log n}{n}}$ such that the (smooth and zero-dimensional affine) scheme with equations

[TABLE]

contains $\mathop{\rm{Spec}}\nolimits{\mathbf{K}}$ as one of its irreducible components.

Theorem 2** (Number fields with bounded discriminant).**

There exists a positive constant ${\mathcal{O}}$ such that the following is true. Let $n\geqslant{\mathcal{O}}$ be an integer. Let $H\geqslant 1$ be an integer. The number of isomorphism classes of number fields with degree $n$ and discriminant $\leqslant H$ is $\leqslant n^{{\mathcal{O}}n\log^{3}n}H^{{\mathcal{O}}\log^{3}n}$.

The meaning of Theorem 1 is that we can describe a number field using few parameters in some sense. We have a short description of it as a quotient of a finite algebra : the smooth zero-dimensional part of a complete intersection of small degree and small height in a projective space of small dimension.

Theorem 2 improves on previous results by Schmidt [17] and Ellenberg-Venkatesh [12]. Schmidt obtains a bound $H^{\frac{n+2}{4}}$ times a function of $n$. Ellenberg and Venkatesh obtain a bound $H^{\exp({\mathcal{O}}\sqrt{\log n})}$ times a function of $n$. We combine techniques from geometry of numbers and interpolation theory to produces small projective models of $\mathop{\rm{Spec}}\nolimits{\mathbf{K}}$ and lower the exponent of $H$ down to ${\mathcal{O}}\log^{3}n$. A key point is to look for local equations rather than a full set of generators of the ideal of these models.

Our estimate is not sharp of course. Indeed for $n=1$ the exponent of $H$ can be taken to be [math]. For $2\leqslant n\leqslant 5$ the exponent of $H$ can be taken to be $1$ according to work by Davenport and Heilbronn [11] for $n=3$, and Bhargava [4, 5] for $n=4,\,5$. It is a bit delicate to infer a general conjecture from these results because the techniques used for these small values of $n$ seem to be quite specific. Cohen, Diaz and Olivier have collected experimental data e.g. in [8, 9, 10] suggesting that the number of isomorphism classes of number fields of degree $n$ and discriminant $\leqslant H$ should grow linearly in $H$ for fixed $n\geqslant 2$. Malle has stated in [14] a more general and accurate conjecture on the distribution of Galois groups of number fields that would confirm this intuition.

In Section 2 we recall notation, definitions and elementary results from the geometry of numbers. In Section 3 we construct models for number fields as irreducible components of complete intersections with small height in low dimensional projective spaces. The last section is devoted to the proof of Theorem 1 and Theorem 2.

The author thanks Pascal Autissier, Karim Belabas, Georges Gras and Christian Maire for their comments and suggestions.

2. Short integers

Let ${\mathbf{K}}$ be a number field and let $n$ be the degree of ${\mathbf{K}}$ over ${\mathbf{Q}}$. Let ${\mathbf{O}}$ be the ring of integers of ${\mathbf{K}}$. Let $(\rho_{i})_{1\leqslant i\leqslant r}$ be the $r$ real embeddings of ${\mathbf{K}}$. Let $(\sigma_{j},\bar{\sigma}_{j})_{1\leqslant j\leqslant s}$ be the $2s$ complex embeddings of ${\mathbf{K}}$. We also denote by $(\tau_{k})_{1\leqslant k\leqslant n}$ the $n=r+2s$ embeddings of ${\mathbf{K}}$. Let

[TABLE]

be the Minkowski space. We follow the presentation in [16, Chapitre 1, §5]. An element $x$ of ${\mathbf{K}}_{\mathbf{R}}$ can be given by $r$ real components $(x_{\rho})_{\rho}$ and $s$ complex components $(x_{\sigma})_{\sigma}$. So we write $x=((x_{\rho})_{\rho},(x_{\sigma})_{\sigma})$. For such an $x$ in ${\mathbf{K}}_{\mathbf{R}}$ we denote by $||x||$ the maximum of the absolute values of its $r+s$ components. The canonical metric on ${\mathbf{K}}_{\mathbf{R}}$ is defined by

[TABLE]

In particular the contribution of complex embeddings is counted twice

[TABLE]

The corresponding Haar measure is said to be canonical also. The canonical measure of the convex body $\{x,||x||\leqslant 1\}$ is

[TABLE]

The map $a\mapsto a\otimes 1$ injects ${\mathbf{K}}$ and ${\mathbf{O}}$ into ${\mathbf{K}}_{\mathbf{R}}$. For every non-zero $x$ in ${\mathbf{O}}$ we have

[TABLE]

Let $(\alpha_{i})_{1\leqslant i\leqslant n}$ be any ${\mathbf{Z}}$-basis of ${\mathbf{O}}$. Set $A=(\tau_{j}(\alpha_{i}))_{1\leqslant i,j\leqslant n}$. The product $A\bar{A}^{t}$ is the Gram matrix $B=(<\alpha_{i},\alpha_{j}>)_{1\leqslant i,j\leqslant n}$ of the canonical form in the basis $(\alpha_{i})_{i}$. This is a real symmetric positive matrix. The volume of ${\mathbf{O}}$ according to the canonical Haar measure is

[TABLE]

The square of the volume of ${\mathbf{O}}$ is the discriminant of ${\mathbf{K}}$

[TABLE]

Applying Minkowski’s second theorem [18, Lecture III, §4, Theorem 16] to the gauge function $x\mapsto||x||$ we find that ${\mathbf{O}}$ contains $n$ linearly independant elements $\omega_{1}$, $\omega_{2}$, …, $\omega_{n}$ such that

[TABLE]

We assume that the sequence $i\mapsto||\omega_{i}||$ is non-decreasing and deduce that

[TABLE]

for every $1\leqslant i\leqslant n$. This inequality is a bit unsatisfactory because it provides little information on the largest $\omega_{i}$. To improve on this estimate we use the fact that ${\mathbf{O}}$ is an integral domain. We let $m=\lceil(n+1)/2\rceil$ be the smallest integer bigger than $n/2$. On the one hand

[TABLE]

for every $1\leqslant i\leqslant m$. On the other hand the products

[TABLE]

generate a ${\mathbf{Z}}$-module of rank $n$. Otherwise there would exist a non-zero linear form $f:{\mathbf{O}}\rightarrow{\mathbf{Z}}$ vanishing on these products. So the $m$ forms $f\circ\omega_{i}$ would be orthogonal to the $m$ vectors $\omega_{j}$. Then $m+m\leqslant n$. A contradiction. We deduce that all the successive minima of ${\mathbf{O}}$ are

[TABLE]

In other words ${\mathbf{O}}$ is well balanced.

Proposition 1** (Number fields have small integers).**

The ring of integers ${\mathbf{O}}$ of a number field ${\mathbf{K}}$ with degree $n$ and discriminant $d_{\mathbf{K}}$ contains $n$ linearly independant elements $(\alpha_{i})_{1\leqslant i\leqslant n}$ over ${\mathbf{Z}}$ such that all the absolute values of all the $\alpha_{i}$ are $\leqslant d_{\mathbf{K}}^{\,2/n}$.

Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and ZhaoSee prove in [3][Theorem 3.1] a similar statement which is somewhat stronger but less accurate.

3. Small models

Let

[TABLE]

Let $d\geqslant 5$ and $r\geqslant 1$ be two integers. We assume that

[TABLE]

Let $M$ be the set of monomials of total degree $\leqslant d$ in the $r$ variables $x_{1}$, …, $x_{r}$. We have

[TABLE]

Let $V_{\mathbf{C}}$ be the ${\mathbf{C}}$-linear space generated by $M$. We may associate to every element in $M$ the corresponding degree $d$ monomial in the $r+1$ variables $x_{0}$, $x_{1}$, …, $x_{r}$. We thus identify $V_{\mathbf{C}}$ with $H^{0}({\mathcal{O}}_{\mathbf{P}^{r}_{\mathbf{C}}}(d))$, the space of homogeneous polynomials of degree $d$.

Let $(P_{\tau})_{\tau}$ be $n$ pairwise distinct points in

[TABLE]

The $P_{\tau}$ are indexed by the $n$ embeddings of ${\mathbf{K}}$. These $n$ points form a set (a reduced zero-dimensional subscheme of $\mathbf{P}^{r}_{\mathbf{C}}$) called $P$. We call ${\mathcal{I}}$ the corresponding ideal sheaf on $\mathbf{P}^{r}_{\mathbf{C}}$. We denote by $2P$ the scheme associated with ${\mathcal{I}}^{2}$. It consists of $n$ double points. We say that the scheme $2P$ is well poised (or non-special) in degree $d$ if it imposes $n(r+1)$ independent conditions on degree $d$ homogeneous polynomials. Equivalently, the map

[TABLE]

is surjective. This is the case if and only if the $n(r+1)\times{d+r\choose d}$ matrix

[TABLE]

has maximal rank $n(r+1)$. We note that ${\mathcal{M}}_{P}^{1}$ consists of $r+1$ blocks of size $n\times{d+r\choose d}$ piled vertically. It has maximal rank for a generic $P$ when $d\geqslant 5$, according to a theorem of Alexander [1], generalized by Alexander and Hirschowitz [2]. Chandler [7, Theorem 1] provides a simpler statement and proof. The recent exposition and simplification by Brambilla and Ottaviani [6] is very useful also.

We now let $(\alpha_{i})_{1\leqslant i\leqslant n}$ be $n$ linearly independant short elements in ${\mathbf{O}}$ as in Proposition 1. We pick $rn$ rational integers $(u_{i,j})_{1\leqslant i\leqslant n,\,1\leqslant j\leqslant r}$ and we set

[TABLE]

for $1\leqslant j\leqslant r$. Let

[TABLE]

be the homomorphism of ${\mathbf{Q}}$-algebras sending $x_{j}$ to $\kappa_{j}$ for $1\leqslant j\leqslant r$. Let

[TABLE]

be the corresponding morphism of schemes. Tensoring $\epsilon_{\mathbf{Q}}$ by ${\mathbf{R}}$ we obtain an homomorphism

[TABLE]

sending $x_{j}$ to $((\rho(\kappa_{j}))_{\rho},(\sigma(\kappa_{j}))_{\sigma})$. We call

[TABLE]

the corresponding morphism of schemes. We define

[TABLE]

and

[TABLE]

similarly. In particular $\epsilon_{\mathbf{C}}$ maps $x_{j}$ onto $(\tau(\kappa_{j}))_{\tau}$.

We now consider the points $(P_{\tau})_{\tau}$ such that $x_{0}(P_{\tau})=1$ and

[TABLE]

for $1\leqslant j\leqslant r$ or equivalently

[TABLE]

The maximal minors of the corresponding matrix ${\mathcal{M}}_{P}^{1}$ are polynomials of total degree $\leqslant dn(r+1)$ in the $u_{i,j}$ and one of them is not identically zero. The latter determinant cannot vanish on the cartesian product $[0,dn(r+1)]^{nr}$. Thus there exist $nr$ rational integers $u_{i,j}$ in the range

[TABLE]

such that the corresponding scheme $2P$ is well poised. We assume that the $u_{i,j}$ meet these conditions.

Since $2P$ is well poised, $P$ is well poised also. So $e_{\mathbf{Q}}$, $e_{\mathbf{R}}$ and $e_{\mathbf{C}}$ are closed embeddings. In order to describe them efficiently we look for polynomials with degree $\leqslant d$ and small integer coefficients vanishing at $P$. We denote by $V_{\mathbf{R}}={\mathbf{R}}[x_{1},\ldots,x_{r}]_{d}$ the ${\mathbf{R}}$-vector space of polynomials in ${\mathbf{R}}[x_{1},\ldots,x_{r}]$ of degree $\leqslant d$. There is a unique ${\mathbf{R}}$-bilinear form on $V_{\mathbf{R}}$ that turns the set $M$ of monomials into an orthonormal basis. The lattice of relations with integer coefficients and degree $\leqslant d$ is the intersection between $\mathop{\rm{Ker}}\nolimits\epsilon_{\mathbf{R}}$ and

[TABLE]

This is a free ${\mathbf{Z}}$-module ${\mathcal{L}}\subset V_{\mathbf{R}}$ of rank

[TABLE]

We set $L={\mathcal{L}}\otimes_{\mathbf{Q}}{\mathbf{R}}$ the underlying ${\mathbf{R}}$-vector space and $L^{\perp}$ its orthogonal complement in $V_{\mathbf{R}}$. We denote by ${\mathcal{L}}^{\perp}$ the intersection ${\mathcal{L}}^{\perp}=L^{\perp}\cap V_{\mathbf{Z}}$. Since $V_{\mathbf{Z}}$ is unimodular, ${\mathcal{L}}$ and ${\mathcal{L}}^{\perp}$ have the same volume. See [15, Corollary 1.3.5.]. We denote by $\hat{\mathbf{O}}=\mathop{\rm{Hom}}\nolimits({\mathbf{O}},{\mathbf{Z}})$ the dual of ${\mathbf{O}}$, the ring of integers of ${\mathbf{K}}$, as a ${\mathbf{Z}}$-module. We call

[TABLE]

the evaluation map in degree $\leqslant d$. We observe that ${\mathcal{L}}^{\perp}$ contains the image of $\hat{\mathbf{O}}$ by the transpose map

[TABLE]

where we have identified ${\mathbf{Z}}[x_{1},\ldots,x_{r}]_{d}$ with its dual thanks to the canonical bilinear form. So the volume of ${\mathcal{L}}$ is bounded from above by the volume of $\hat{\epsilon}_{{\mathbf{Z}},d}(\hat{\mathbf{O}})$. We consider the matrix

[TABLE]

of the map $\epsilon_{{\mathbf{C}},d}=\epsilon_{{\mathbf{Z}},d}\otimes_{\mathbf{Z}}{\mathbf{C}}$ in the canonical bases. If we prefer to use an integral basis of ${\mathbf{O}}$ on the right we should multiply ${\mathcal{M}}_{P}^{0}$ on the left by the inverse $T$ of the matrix of a basis of ${\mathbf{O}}$ in the canonical basis. We deduce that the square of the volume of $\hat{\epsilon}_{{\mathbf{Z}},d}(\hat{\mathbf{O}})$ is the determinant of $T{\mathcal{M}}_{P}^{0}({\mathcal{M}}_{P}^{0})^{t}T^{t}$. Since $T{\mathcal{M}}_{P}^{0}$ has real coefficients we have

[TABLE]

So the square of the volume of the lattice of relations is bounded by the determinant of the hermitian positive definite matrix ${\mathcal{M}}_{P}^{0}\left(\overline{{\mathcal{M}}_{P}^{0}}\right)^{t}$ divided by $d_{\mathbf{K}}$.

Recall that the coefficients in ${\mathcal{M}}_{P}^{0}$ are degree $\leqslant d$ monomials in the $\kappa_{j}=\sum_{1\leqslant i\leqslant n}u_{i,j}\alpha_{i}$. The coefficients $u_{i,j}$ are bounded form above by $dn(r+1)$. All the absolute values of the $\alpha_{i}$ are bounded from above by $d_{\mathbf{K}}^{2/n}$. So the coefficients in ${\mathcal{M}}_{P}^{0}$ are bounded from above by

[TABLE]

The coefficients in ${\mathcal{M}}_{P}^{0}\left(\overline{{\mathcal{M}}_{P}^{0}}\right)^{t}$ are bounded from above by

[TABLE]

The matrix ${\mathcal{M}}_{P}^{0}\left(\overline{{\mathcal{M}}_{P}^{0}}\right)^{t}$ being hermitian positive definite, its determinant is bounded from above by the product of the diagonal terms. We deduce that the volume of the lattice ${\mathcal{L}}$ of relations is bounded from above by ${\mathfrak{D}}^{n/2}$. Recall that the dimension of ${\mathcal{L}}$ is

[TABLE]

For any $x$ in $V_{\mathbf{R}}$ we denote by $||x||$ the $\ell_{2}$-norm in the monomial basis. The volume of the sphere $\{x\in L,||x||\leqslant 1\}$ is $\geqslant 2^{\ell}\ell^{-\ell/2}$. Applying Minkowski’s second theorem [18, Lecture III, §4, Theorem 16] to the gauge function $x\mapsto||x||$ we find that ${\mathcal{L}}$ contains $\ell$ linearly independant elements $E_{1}$, $E_{2}$, …, $E_{\ell}$ such that

[TABLE]

We assume that the sequence $i\mapsto||E_{i}||$ is non-decreasing and deduce that the size of the $i$-th equation is bounded from above

[TABLE]

for every $1\leqslant i\leqslant\ell$. Again, this inequality is a bit unsatisfactory because it provides little information on the largest equations. This time we see no other way around than forgetting the last $n-1$ equations. On the one hand

[TABLE]

for every $1\leqslant i\leqslant\ell+1-n$.

On the other hand the scheme $2P$ is well poised and the ${\mathbf{C}}$-vector space generated by the $E_{i}$ for $1\leqslant i\leqslant\ell+1-n$ has codimension $n-1<n$ in $L\otimes_{\mathbf{R}}{\mathbf{C}}$. So there exists at least one embedding $\tau$ such that the $(\ell+1-n)\times r$ matrix

[TABLE]

has maximal rank $r$. In more geometric terms the ${\mathbf{C}}$-vector space generated by the $\ell+1-n$ first equations $(E_{i})_{1\leqslant i\leqslant\ell+1-n}$ surjects onto the cotangent space to $\mathbf{P}^{r}_{\mathbf{C}}$ at the geometric point $P_{\tau}$ for at least one $\tau$. This means that there exist $r$ integers $1\leqslant i_{1}<i_{2}<\dots<i_{r}\leqslant\ell+1-n$ such that the minor determinant

[TABLE]

is non-zero for some $\tau$ and thus for all $\tau$ by Galois action.

Proposition 2** (Number fields have small models).**

Let ${\mathbf{K}}$ be a number field of degree $n$ and discriminant $d_{\mathbf{K}}$ over ${\mathbf{Q}}$. Let $d\geqslant 5$ and $r\geqslant 1$ be rational integers such that

[TABLE]

There exists $r$ polynomials $E_{1}$, $E_{2}$, …, $E_{r}$ of degree $\leqslant d$ in ${\mathbf{Z}}[x_{1},\ldots,x_{r}]$ having coefficients bounded in absolute value by

[TABLE]

where

[TABLE]

and such that the (smooth and zero-dimensional affine) scheme with equations

[TABLE]

contains $\mathop{\rm{Spec}}\nolimits{\mathbf{K}}$ as one of its irreducible components.

4. Proof of main results

In this section, the notation ${\mathcal{O}}$ stands for a positive absolute constant. Any sentence containing this symbol becomes true if the symbol is replaced in every occurrence by some large enough real number.

We specialize the values of the parameters $r$ and $d$ in Proposition 2. We will take $d=r$. It is evident that ${2r\choose r}\geqslant 2^{r}$ so

[TABLE]

for $r$ large enough. Further

[TABLE]

We choose $r$ to be the smallest positive integer such that $n(r+1)\leqslant{2r\choose r}$. We have

[TABLE]

for $n$ large enough. We deduce that $\ell={2r\choose r}-n\leqslant 4n(r+1)\leqslant{\mathcal{O}}n\log n$. So

[TABLE]

From Equation (1) we deduce that ${2r\choose r}\leqslant{\mathcal{O}}n\log n$. Also $n^{2}d(r+1)\leqslant{\mathcal{O}}n^{2}\log^{2}n$ and

[TABLE]

So the coefficients of equations $E_{i}$ are bounded in absolute value by

[TABLE]

This proves Theorem 1. Theorem 2 follows because there are $r{2r\choose r}$ coefficients to be fixed. We note also that there may appear several number fields in the smooth zero dimensional part of the complete intersection $E_{1}=E_{2}=\dots=E_{r}=0$. However the Chow class of this intersection is $r^{r}\leqslant(\log n)^{{\mathcal{O}}\log n}$ and the number of isolated points is bounded by this intersection number [13, Chapter 13].