Systems of cubic forms in many variables

TL;DR

This paper proves an asymptotic formula for counting integer solutions to systems of smooth cubic forms in many variables, significantly reducing the variable count needed compared to previous results.

Contribution

It establishes an asymptotic count for solutions when the number of variables is at least 25 times the number of forms, improving upon the prior requirement of much larger variable counts.

Findings

Asymptotic formula for solutions in systems with n ≥ 25R variables.

Reduction of variable count needed from R^2 to linear in R.

Application of Davenport's method to bound solutions of auxiliary inequalities.

Abstract

We consider a system of $R$ cubic forms in $n$ variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided $n\geq 25R$, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular we can handle systems of forms in $O(R)$ variables, previous work having required that $n \gg R^2$. One conjectures that $n \geq 6R+1$ should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport.

Equations

$$c_1(\mathbf{x})=0,\dots ,c_R(\mathbf{x})=0$$

$$N_{c_1,\dots ,c_R}(P)=\#\{\mathbf{x}\in {\mathbb{Z}}^n:\mathbf{x}/P\in \mathcal{B},c_1(\mathbf{x})=0,\dots ,c_R(\mathbf{x})=0\}.$$

$$n\ge 8R^2+9R,$$

$$N_{c_1,\dots ,c_R}(P)=\mathfrak{IS}{P}^{n-3R}+O(P^{n-3R-\delta })$$

$$n\ge 25R.$$

$$H_{c}(\bm{x})=\frac{1}{\lVert c\rVert_{\infty}}\bigg{(}\frac{\partial^{3}c(\bm{x})}{\partial x_{i}\partial x_{j}}\bigg{)}_{1\leq i,j\leq n}$$

$$\Vert \mathbf{x}{\Vert }_{\infty },\Vert \mathbf{y}{\Vert }_{\infty }\le B,\Vert H_c(\mathbf{x})\mathbf{y}{\Vert }_{\infty }<B.$$

$${\mathbf{m}}^{(f)}({\mathbf{x}}^{(1)},{\mathbf{x}}^{(2)})=\Vert c{\Vert }_{\infty }{H}_c({\mathbf{x}}^{(1)}){\mathbf{x}}^{(2)}.$$

$$N_{\mathbf{\beta }\cdot \mathbf{c}}^{\mathrm{aux}}(B)\le C_0{B}^{2n-8\mathcal{C}}$$

$$N_{c_1,\dots ,c_R}(P)=\mathfrak{IS}{P}^{n-dR}+O(P^{n-dR-\delta })$$

$${\sigma }_{\mathcal{K}}=1+\underset{c\in \mathcal{K}\setminus \{0\}}{\max }\dim \mathrm{Sing}V(c),$$

$$N_c^{\mathrm{aux}}(B){\ll }_{\mathcal{K},ϵ}{B}^{n+{\sigma }_{\mathcal{K}}+ϵ}.$$

$$N_{\mathbf{\beta }\cdot \mathbf{c}}^{\mathrm{aux}}(B){\ll }_{c_1,\dots ,c_R,ϵ}{B}^{n+R-1+ϵ}$$

$$\dim \mathrm{Sing}V(\mathbf{\beta }\cdot \mathbf{c})\ge R-1.$$

$$V(c_1,\dots ,c_{R-1})\cap \mathrm{Sing}V(\mathbf{\beta }\cdot \mathbf{c})\subset \mathrm{Sing}V(c_1,\dots ,c_R),$$

$$N_c^{aux-eq}(B)=\#\{(\mathbf{x},\mathbf{y})\in ({\mathbb{Z}}^n{)}^2:\Vert \mathbf{x}{\Vert }_{\infty },\Vert \mathbf{y}{\Vert }_{\infty }\le B,H_c(\mathbf{x})\mathbf{y}=0\}$$

$$|{\lambda }_{c,1}(\mathbf{x})|\le n\Vert H_c(\mathbf{x}){\Vert }_{\infty }\le n^2\Vert \mathbf{x}{\Vert }_{\infty }.$$

$$N_H(B)=\#\{\mathbf{y}\in {\mathbb{Z}}^n:\Vert \mathbf{y}{\Vert }_{\infty }\le B,\Vert H\mathbf{y}{\Vert }_{\infty }\le B\}.$$

$$N_H(B){\ll }_{C,n}\underset{1\le i\le n}{\min }\frac{B^n}{1+|{\lambda }_1\cdots {\lambda }_i|}.$$

$$\{\mathbf{t}\in {\mathbb{R}}^n:{\mathbf{t}}^T{H}^{T}H\mathbf{t}\le n{B}^2\},$$

$$N_H(B)$$

$$\le \prod _{i=1}^n\min \{2C|{\lambda }_i{|}^{-1}\sqrt{n}B,B\}.$$

$$N_H(B){\ll }_{C,n}{B}^n\prod _{i=1}^n\min \{|{\lambda }_i{|}^{-1},1\}.$$

$$N_H(B)$$

$$\ll \underset{1\le i\le n}{\min }\frac{B^n}{1+|{\lambda }_1\cdots {\lambda }_i|}.\text{qed}$$

$$\frac{1}{2}{E}_i<|{\lambda }_{c,i}(\mathbf{x})|\le E_i$$

$$|{\lambda }_{c,i}(\mathbf{x})|\le E_{k+1}$$

$$\frac{N_c^{\mathrm{aux}}(B)}{B^n(\log B{)}^n}{\ll }_n\#\{{\mathbb{Z}}^n\cap K_0(1)\},$$

$$\frac{2^{e_{1}+\dotsb+e_{k}}N^{\operatorname{aux}}_{c}(B)}{B^{n}(\log B)^{n}}\ll_{n}\#\big{\{}\mathbb{Z}^{n}\cap K_{k}(2^{e_{1}},\dotsc,2^{e_{k}},1)\big{\}},$$

$$\frac{2^{e_{1}+\dotsb+e_{n}}N^{\operatorname{aux}}_{c}(B)}{B^{n}(\log B)^{n}}\ll_{n}\#\big{\{}\mathbb{Z}^{n}\cap K_{n-1}(2^{e_{1}},\dotsc,2^{e_{n}})\big{\}}.$$

$$K_n(2^{e_1},\dots ,2^{e_n},1)\subset K_{n-1}(2^{e_1},\dots ,2^{e_n}).$$

Full text

1. 1 Introduction

2. 1.1 Main result

3. 1.2 Related work

4. 1.3 Reduction to an auxiliary inequality

5. 1.4 Outline of remaining steps

6. 1.5 Structure of this paper

7. 1.6 Notation

8. 2 The eigenvalues of the Hessian matrix $H_{c}(\bm{x})$

9. 3 Intermission: Eigenvalues and minors

10. 4 Counting points in the sets $K_{k}(E_{1},\dotsc,E_{k+1})$

11. 5 Small values of a trilinear form

12. 6 Constructing singular points on $V(c)$

13. 7 Acknowledgements

1 Introduction

1.1 Main result

Let $c_{1}(\bm{x}),\dotsc,c_{R}(\bm{x})$ be homogeneous cubic forms in $n$ variables $x_{1},\dotsc,x_{n}$, with integer coefficients. We treat the simultaneous Diophantine equations

[TABLE]

and the corresponding projective variety in $\mathbb{P}_{\mathbb{Q}}^{n-1},$ which we call $V(c_{1},\dots,c_{R})$. We assume throughout that the $c_{i}$ generate the ideal of $V(c_{1},\dots,c_{R}),$ and are linearly independent. The cubic case of a classic result of Birch gives us:

Theorem 1.1** (Birch [2]).**

Let $\mathscr{B}$ be a box in $\mathbb{R}^{n},$ contained in the box $[-1,1]^{R},$ and having sides of length at most 1 which are parallel to the coordinate axes. For each $P\geq 1,$ write

[TABLE]

If the variety $V(c_{1},\dots,c_{R})$ is a smooth complete intersection, and we have

[TABLE]

then for some $\mathfrak{I},$ $\mathfrak{S}\geq 0$ and all $P\geq 1,$ the bound

[TABLE]

holds, where the implicit constant depends only on the forms $c_{i},$ and $\delta$ is a positive real number depending only on $d$ and $R$. If the variety $V(c_{1},\dotsc,c_{R})$ has a smooth point over $\mathbb{Q}_{p}$ for each prime $p,$ and a real point whose homogeneous co-ordinates lie in $\mathscr{B},$ then $\mathfrak{S}$ and $\mathfrak{I}$ are positive.

In particular this follows from Theorem 1 of Birch [2], on inserting the bound $\dim V^{\ast}\leq R-1$ for the dimension of the variety $V^{\ast}$ occurring in that result. This bound follows from Lemma 3.1 of Browning and Heath-Brown [4] whenever $V(c_{1},\dotsc,c_{R})$ is a smooth complete intersection. See [17, Lemma LABEL:1.lem:nonsing_case] for details.

We sharpen (1.1) as soon as $R\geq 3$. In §1.3 we prove:

Theorem 1.2**.**

In Theorem 1.1 we may replace (1.1) with the condition

[TABLE]

For example when $R=3$ and $V(c_{1},c_{2},c_{3})$ is a smooth complete intersection, Theorem 1.2 applies when $n\geq 75,$ whereas Birch’s theorem requires $n\geq 99$.

The “square-root cancellation” heuristic suggests that in place of (1.1) the condition $n\geq 6R+1$ should suffice, see for example the discussion around formula (1.5) in Browning [3]. By handling systems of forms in $O(R)$ variables we come within a constant factor of this conjecture.

Our strategy is an extension of our previous work [17]. In forthcoming papers we further generalise this approach to treat systems of $R$ forms with degree $d\geq 2,$ with rational or real coefficients.

1.2 Related work

We begin with the case when the forms $c_{i}(\bm{x})$ are diagonal.

In the case of a single diagonal form $c,$ Baker [1] proves that $V(c)$ has a rational point whenever $n\geq 7$.

Brüdern and Wooley [8, 7, 11] treat diagonal systems in $n\geq 6R+1$ variables. This is the best value of $n$ possible with the classical circle method. In particular they prove the Hasse principle for $V(c_{1},\dots,c_{R})$ whenever the $c_{i}$ are diagonal, $V(c_{1},\dots,c_{R})$ is smooth and $n\geq 6R+1$. They also prove an asymptotic formula of the type (1.2) whenever $n\geq 6R+3$ holds, or when $R=2$ and $n\geq 14$ holds [5, 6, 9]. In the case $R=2$ they prove a Hasse principle for certain pairs of diagonal cubics in as few as 11 variables [10].

Returning to the case of general (not necessarily diagonal) forms, we consider the case $R=1$. Let $c$ be a cubic form. Hooley [16] proves that if $n=8,$ the variety $V(F)$ is smooth, and the box $\mathscr{B}$ does not contain a point at which the Hessian determinant of $F$ vanishes, then the asymptotic formula (1.2) holds. In this work he assumes a Riemann hypothesis for a certain modified Hasse-Weil $L$-function. When $n=9$ he proves the same result unconditionally, the weaker error term $o(P^{n-3})$ in place of the $O(P^{n-3-\delta})$ from (1.2). Heath-Brown [15] proves that if $n\geq 14$ then $V(c)$ always has a rational point, regardless of whether it is singular.

In the case $R=2,$ Dietmann and Wooley [14] have shown that $V(c_{1},c_{2})$ always has a rational point when $n\geq 827,$ whether or not it is smooth.

In the general case $R\geq 1,$ Schmidt [18] shows that $V(c_{1},\dots,c_{R})$ always has a rational point if $n\geq(10R)^{5}$. Recent work of Dietmann [13] improves this condition to $n\geq 400~{}000R^{4}$.

1.3 Reduction to an auxiliary inequality

To prove Theorem 1.2 we will use Theorem LABEL:1.thm:manin from the author’s previous work [17]. This will reduce the problem to proving an upper bound for the number of solutions to the following auxiliary inequality.

Definition 1.1**.**

For any $k\geq 1$ and $\bm{t}\in\mathbb{R}^{k},$ we write $\lVert\bm{t}\rVert_{\infty}=\max_{i}\lvert t_{i}\rvert$ for the supremum norm. When $c(\bm{x})$ is a real cubic form in $n$ variables with real coefficients, we define a symmetric matrix

[TABLE]

where \lVert c\rVert_{\infty}=\frac{1}{6}\max_{i,j,k\in\{1,\dotsc,n\}}\big{\lvert}\frac{\partial^{3}c(\bm{x})}{\partial x_{i}x_{j}x_{k}}\big{\rvert}. Thus $H_{c}(\bm{x})$ is the Hessian of the cubic form $c(\bm{x})/\lVert c\rVert_{\infty},$ which has been normalised so that 1 is the absolute value of its largest coefficient. For each $B\geq 1$ we put $N^{\operatorname{aux}}_{c}(B)$ for the number of pairs of vectors $(\bm{x},\bm{y})\in(\mathbb{Z}^{n})^{2}$ with

[TABLE]

We show that this definition of $N^{\operatorname{aux}}_{c}(B)$ above agrees with the one given in [17, Definition LABEL:1.def:aux_ineq]. There we consider a degree $d$ polynomial $f$ and a system of multilinear forms $\bm{m}^{(f)}(\bm{x}^{(1)},\dotsc,\bm{x}^{(d-1)}),$ and when $d=3$ and $f(\bm{x})=c(\bm{x}),$ we see that

[TABLE]

It follows that the definition above agrees with Definition LABEL:1.def:aux_ineq from [17]. The case $d=3$ of Theorem LABEL:1.thm:manin in [17] therefore states that:

Theorem 1.3**.**

Let the counting function $N_{c_{1},\dotsc,c_{R}}(P)$ be as in Theorem 1.1. Suppose that for some $C_{0}\geq 1$ and ${\mathscr{C}}>3R,$ we have

[TABLE]

for all $\bm{\beta}\in\mathbb{R}^{R}$ and $B\geq 1,$ where we write $\bm{\beta}\cdot\bm{c}$ for $\beta_{1}c_{1}+\dotsb+\beta_{R}c_{R}$. Then for some $\mathfrak{I},$ $\mathfrak{S}\geq 0$ we have

[TABLE]

for all $P\geq 1,$ where the implicit constant depends at most on $C_{0},$ ${\mathscr{C}}$ and the $c_{i},$ and the positive constant $\delta$ depends at most on ${\mathscr{C}},$ $d$ and $R$. The constants $\mathfrak{I}$ and $\mathfrak{S}$ are positive under the same conditions as in Theorem 1.1.

We give the following bound for the counting function $N^{\operatorname{aux}}_{c}(B)$. The proof occupies the bulk of this paper and is completed in §6.

Proposition 1.1**.**

We call a set ${\mathcal{K}}$ of real cubic forms in $n$ variables a closed cone if (i) for all $c\in{\mathcal{K}}$ and $\lambda\geq 0$ we have $\lambda c\in K,$ and (ii) ${\mathcal{K}}$ is closed in the real linear space of cubic forms in $n$ variables.

Let ${\mathcal{K}}$ be a closed cone as above, and let $N^{\operatorname{aux}}_{c}(B)$ be as in Definition 1.1. If we set

[TABLE]

so that $\sigma_{\mathcal{K}}\in\{0,\dotsc,n-1\},$ then for all $\epsilon>0,$ $c\in{\mathcal{K}}$ and $B\geq 1$ we have

[TABLE]

We will outline the proof after deducing Theorem 1.2.

Proof of Theorem 1.2.

Suppose that (1.3) holds. We claim that for all $B\geq 1,$ $\epsilon>0$ and $\bm{\beta}\in\mathbb{R}^{R}$ we have

[TABLE]

where $\bm{\beta}\cdot\bm{c}$ is as in Theorem 1.3. If we set ${\mathscr{C}}=(n-R+\frac{1}{2})/8$ and let $C_{0}$ be sufficiently large in terms of the forms $c_{i},$ we can then apply Theorem 1.3. For (1.8) implies (1.5) on setting $\epsilon=\frac{1}{2}$ in (1.8). Moreover we have ${\mathscr{C}}>3R,$ by (1.3). So the hypotheses of Theorem 1.3 are satisfied, and Theorem 1.2 follows.

Setting ${\mathcal{K}}=\{\bm{\beta}\cdot\bm{c}:\bm{\beta}\in\mathbb{R}^{R}\}$ in Proposition 1.1, we see that (1.8) follows from (1.7) unless $\sigma_{\mathcal{K}}>R-1$ holds. Suppose for a contradiction that we have $\sigma_{\mathcal{K}}>R-1$.

By the definition (1.6) there must be $\bm{\beta}\in\mathbb{R}^{R}\setminus\{\bm{0}\}$ with

[TABLE]

We may assume that $V(c_{1},\dots,c_{R})=V(c_{1},\dotsc,c_{R-1},\bm{\beta}\cdot\bm{c})$ holds, after permuting the $c_{i}$ if necessary. Since $V(c_{1},\dots,c_{R})$ is a smooth complete intersection, we then have

[TABLE]

and so $\dim\operatorname{Sing}V(c_{1},\dots,c_{R})>\dim\operatorname{Sing}V(\bm{\beta}\cdot\bm{c})-R$ holds. Thus (1.9) implies that $V(c_{1},\dots,c_{R})$ is singular, which is false by assumption. ∎

1.4 Outline of remaining steps

To prove Proposition 1.1 we adapt the argument used to prove Lemma 3 in Davenport [12], and subsequently a somewhat more general result in §5 of Schmidt [18]. These authors consider the counting function defined by

[TABLE]

for some cubic form $c$ with integer coefficients. Davenport proves that either $N^{\operatorname{aux-eq}}_{c}(B)$ is small, or there is a large rational linear space on which $c$ vanishes. In order to briefly sketch his line of reasoning, we define some additional notation.

Definition 1.2**.**

Let $\lVert H_{c}(\bm{x})\rVert_{\infty}=\max_{i,j}\lvert H_{c}(\bm{x})_{ij}\rvert$ and let ${\lambda_{c,1}(\bm{x})},\dotsc,{\lambda_{c,n}(\bm{x})}$ be the eigenvalues of the real symmetric matrix $H_{c}(\bm{x}),$ listed with multiplicity and in order of decreasing absolute value. Observe that

[TABLE]

For each $i\in\{1,\dotsc,n\}$ let $\bm{D}^{(c,i)}(\bm{x})$ be the vector of all $i\times i$ minors of $H_{c}(\bm{x}),$ arranged in some order. This is a vector of degree $i$ homogeneous forms in the variables $\bm{x},$ with real coefficients. Let $J_{\bm{D}^{(c,i)}}(\bm{x})$ be the Jacobian matrix $(\partial\Delta^{(c,i)}_{j}(\bm{x})/\partial x_{k})_{jk}$.

Davenport’s argument runs as follows.

1. (1)

Let $\sigma\in\{0,\dotsc,n-1\}$. Suppose that $N^{\operatorname{aux-eq}}_{c}(B)\gg B^{n+\sigma}$ for some sufficiently large implicit constant. The contribution to this count from any one vector $\bm{x}$ is at most $O(B^{n-\operatorname{rank}H_{c}(\bm{x})})$. So there must be an integer $b$ in the set $\{0,\dotsc,n-1\}$ such that at least $\gg B^{\sigma+b}$ integer points $\bm{x}$ satisfy both $\operatorname{rank}H_{c}(\bm{x})=b$ and $\lVert\bm{x}\rVert_{\infty}\leq B$. 2. (2)

If $\sigma,b$ are as in 1, then it follows that there is an integer point $\bm{x}^{(0)}$ such that $\operatorname{rank}H_{c}(\bm{x}^{(0)})=b$ holds and the tangent space to the affine variety $\bm{D}^{(c,b+1)}(\bm{x})=\bm{0}$ at the point $\bm{x}^{(0)}$ has dimension $\sigma+b+1$ or more. Equivalently, $\operatorname{rank}H_{c}(\bm{x}^{(0)})=b$ and $\operatorname{rank}J_{\bm{D}^{(c,b+1)}}(\bm{x}^{(0)})\leq n-\sigma-b-1$ both hold. This follows from Lemma 2 of Davenport [12]. 3. (3)

If there exists a vector $\bm{x}^{(0)}$ as in 2, then it follows that there exist linear subspaces $X,Y$ of $\mathbb{Q}^{n},$ with dimensions $\sigma+b+1$ and $n-b$ respectively, such that for all $\bm{X}\in X$ and $\bm{Y},\bm{Y}^{\prime}\in Y$ the equality $\bm{Y}^{T}H_{c}(\bm{X})\bm{Y}^{\prime}=0$ holds. See Lemma 4 in Schmidt [18] or the proof of Lemma 3 in Davenport [12]. 4. (4)

We conclude that if $N^{\operatorname{aux-eq}}_{c}(B)\gg B^{n+\sigma}$ then there are spaces $X,Y$ as in 3. So the space $Z$ defined by $Z=X\cap Y$ is a rational linear space, with dimension at least $\sigma+1,$ such that for all $\bm{Z}\in Z$ the equality $c(\bm{Z})=0$ holds.

Our setting differs in three ways from that of Schmidt and Davenport. First, we consider the inequality $\lVert H_{c}(\bm{x})\bm{y}\rVert_{\infty}\leq B$ rather than the equation $H_{c}(\bm{x})\bm{y}=\bm{0}$. Second, for us the cubic form $c(\bm{x})$ may have real coefficients. And third, rather than concluding that $c(\bm{x})$ has a rational linear space of zeroes, we seek to show that the variety $V(c)$ is very singular.

1.5 Structure of this paper

In §2 and the three sections §§4-6 we will modify each of the four steps 1-4 above to accommodate the three changes described at the end of §1.4. In the remaining section, §3, we prove some technical lemmas relating the minors and eigenvalues of real matrices.

1.6 Notation

Throughout, we let $c,$ $\lVert\bm{t}\rVert_{\infty},$ $\lVert c\rVert_{\infty},$ $H_{c}(\bm{x})$ and $N^{\operatorname{aux}}_{c}(B)$ be as in Definition 1.1, and we let $\lVert H_{c}(\bm{x})\rVert_{\infty},$ $\lambda_{c,i}(\bm{x}),$ $\bm{D}^{(c,i)}(\bm{x})$ and $J_{\bm{D}^{(c,i)}}(\bm{x})$ be as in Definition 1.2. We do not require algebraic varieties to be irreducible, and we adopt the convention that $\dim\emptyset=-1$. We use Vinogradov’s $\ll$ notation and big-$O$ notation in the usual way.

2 The eigenvalues of the Hessian matrix $H_{c}(\bm{x})$

We show that if the counting function $N^{\operatorname{aux}}_{c}(B)$ from Definition 1.1 is large, then there are many integer points $\bm{x}$ for which the eigenvalues of $H_{c}(\bm{x})$ lie in some fixed dyadic ranges. This corresponds to step 1 from §1.4.

Lemma 2.1**.**

Let $H$ be a real symmetric $n\times n$ matrix and let ${\lambda_{1}},\dotsc,{\lambda_{n}}$ be the eigenvalues of the matrix $H,$ listed with multiplicity and in order of decreasing absolute value. Let $C\geq 1$ and $B\geq 1,$ and suppose that $\lvert\lambda_{1}\rvert\leq CB$ holds. Set

[TABLE]

Then we have

[TABLE]

Proof.

The integral vectors $\bm{y}$ counted by $N_{H}(B)$ are all contained in the box $\lVert\bm{y}\rVert_{\infty}\leq B,$ and in the ellipsoid

[TABLE]

which has principal radii $\lvert\lambda_{i}\rvert^{-1}\sqrt{n}B$. Hence

[TABLE]

It follows that

[TABLE]

Since the inequalities $\lvert\lambda_{1}\rvert\geq\dotsb\geq\lvert\lambda_{n}\rvert$ hold, we deduce that

[TABLE]

Definition 2.1**.**

Suppose that $k\in\{0,\dotsc,n\}$ and that $E_{1},\dotsc,E_{k+1}\in\mathbb{R}$ such that the inequalities $E_{1}\geq\dotsb\geq E_{k+1}\geq 1$ hold. Then we define $K_{k}(E_{1},\dotsc,E_{k+1})$ to be the set of all vectors $\bm{x}$ in $\mathbb{R}^{n}$ satisfying the following conditions: the inequality $\lVert\bm{x}\rVert_{\infty}\leq B$ holds, and we have

[TABLE]

whenever $1\leq i\leq k$ holds, and we have

[TABLE]

whenever $k+1\leq i\leq n$ holds.

Corollary 2.1**.**

Let $N^{\operatorname{aux}}_{c}(B)$ be as in Definition 1.1, let $\lambda_{c,i}(\bm{x})$ and $\bm{D}^{(c,i)}(\bm{x})$ be as in Definition 1.2, and let $K_{k}(E_{1},\dotsc,E_{k+1})$ be as in Definition 2.1. For any $B\geq 1,$ one of the following alternatives holds. Either

[TABLE]

or there is $k\in\{1,\dotsc,n-1\}$ and there are $e_{1},\dotsc,e_{k}\in\mathbb{N}$ such that the inequalities $\log B\gg_{n}e_{1}\geq\dotsb\geq e_{k}$ hold and

[TABLE]

or there are $e_{1},\dotsc,e_{n}\in\mathbb{N}$ satisfying $\log B\gg_{n}e_{1}\geq\dotsb\geq e_{n}$ and

[TABLE]

Proof.

Note that in the case that $k=n,$ there are no values of $i$ satisfying $k+1\leq i\leq n,$ so the last condition in the definition of $K_{k}(E_{1},\dotsc,E_{k+1})$ is vacuously true and can be omitted. In particular, if $k=n$ then (2.3) follows from (2.2), because

[TABLE]

So it is enough to prove that either (2.1) holds or there exist integers $k$ and $e_{1},\dotsc,e_{k},$ satisfying the inequalities $1\leq k\leq n$ and $\log B\gg_{n}e_{1}\geq\dotsb\geq e_{n},$ such that (2.2) holds.

Now the set $K_{0}(1),$ together with the sets $K_{k}(2^{e_{1}},\dotsc,2^{e_{k}},1),$ partition the box $\lVert\bm{x}\rVert_{\infty}\leq B$ into disjoint pieces. So, if we let

[TABLE]

then we have

[TABLE]

The total number of terms on the right-hand side of (2.4) is $O_{n}((\log B)^{n})$ at most, so it follows that either

[TABLE]

holds, or else there are $1\leq k\leq n$ and $e_{1}\geq\dotsb\geq e_{k}\geq 1$ such that

[TABLE]

If (2.5) holds then the trivial bound $N_{H_{c}(\bm{x})}(B)\ll_{n}B^{n}$ implies (2.1). Suppose instead that (2.6) holds.

By (1.10), for each real vector $\bm{x}$ the bound $\lvert\lambda_{c,1}(\bm{x})\rvert\ll_{n}B$ holds. So we may apply Lemma 2.1 with the choice $H=H_{c}(\bm{x})$ and some $C$ depending on $n$ only. This shows that

[TABLE]

Substituting this into (2.6) we see that (2.2) holds, as claimed. ∎

3 Intermission: Eigenvalues and minors

Here we collect some elementary facts about the eigenvalues and minors of real matrices which will be needed in §§4-5.

Lemma 3.1**.**

For each $k,\ell\in\mathbb{N},$ let

[TABLE]

This set has ${\ell\choose k}$ members. For each $k,\ell,m\in\mathbb{N}$ such that $k\leq\min\{\ell,m\},$ and each $\ell\times m$ real matrix $L,$ define an ${\ell\choose k}\times{m\choose k}$ real matrix $L^{[k]}$ by

[TABLE]

so that the $L^{[k]}_{\bm{a}\bm{b}}$ are the $k\times k$ minors of $L$. For all $\ell\times m$ matrices $L,$ all $m\times n$ matrices $M$ and all $k\leq\min\{\ell,m,n\}$ we have $(LM)^{[k]}=L^{[k]}M^{[k]}$. That is, we have

[TABLE]

Proof.

Let $\bm{e}^{(1)},\dotsc,\bm{e}^{(m)}$ be the standard basis of $\mathbb{R}^{m}$. Fix $L,\bm{a},\bm{b}$; then each side of (3.1) is an alternating multilinear form in those $k$ columns of $M$ whose indices appear in the vector $\bm{b}$. This is some $k$-tuple of $m$-vectors.

If one is given the value of an alternating multilinear form at the $k$-tuple $\bm{e}^{(z_{1})},\dotsc,\bm{e}^{(z_{k})}$ for each $\bm{z}\in T_{k,m},$ one can extend by linearity and the alternating property to find its value at any $k$-tuple of $m$-vectors. In other words, it suffices to check (3.1) when, for some $\bm{z}\in T_{k,m},$ the $k\times k$ submatrix $(M_{z_{i}b_{j}})_{1\leq i,j\leq k}$ is the identity and all other entries of $M$ are zero. In this case both sides of (3.1) are equal to $M^{[k]}_{\bm{z}\bm{b}}$. ∎

Lemma 3.2**.**

Let $M$ be a real $m\times n$ matrix. Recall that $M^{T}M$ is positive semidefinite and symmetric. Let the eigenvalues of $M^{T}M$ be $\Lambda_{1}^{2},\dotsb,\Lambda_{n}^{2}$ in decreasing order, where the $\Lambda_{i}$ are nonnegative and in decreasing order. That is, the $\Lambda_{i}$ are the singular values of $M,$ listed in decreasing order.

In particular, if $M$ is a symmetric matrix, then the $\Lambda_{i}$ are exactly the absolute values of the eigenvalues of $M,$ by diagonalisation.

Given a natural number $k$ with $k\leq\min(m,n),$ let $\bm{D}^{(k)}$ be the vector of $k\times k$ minors of $M,$ arranged in some order. Then we have:

1. (i)

The maximum norm $\lVert\bm{\Delta}^{(k)}\rVert_{\infty}$ satisfies

[TABLE] 2. (ii)

There is a $k$-dimensional linear space $V\subset\mathbb{R}^{n}$ such that for all $\bm{v}\in V$,

[TABLE]

We may take $V$ to be a span of $k$ standard basis vectors $\bm{e}^{(i)}$ in $\mathbb{R}^{n}$. 3. (iii)

For any $C\geq 1,$ either there is an $(n-k+1)$-dimensional linear subspace $X$ of $\mathbb{R}^{n}$ such that

[TABLE]

or there is a $k$-dimensional linear subspace $V$ of $\mathbb{R}^{n},$ spanned by standard basis vectors of $\mathbb{R}^{n},$ such that

[TABLE]

Proof. Part i.

First we prove the result on the assumption that $M^{T}M$ is diagonal. Let the sets $T_{k,\ell}$ and the matrices $L^{[k]}$ be as in Lemma 3.1. Since $M^{T}M$ is diagonal with diagonal entries $\Lambda_{i}^{2},$ we have

[TABLE]

by (3.1). The left-hand side of (3.5) is $\asymp_{n}\Lambda_{1}^{2}\dotsm\Lambda_{k}^{2},$ and the right-hand side is $\asymp_{m,n}\lVert\bm{\Delta}^{(k)}\rVert_{\infty}^{2},$ so this proves (3.2).

Let $O$ be an $n\times n$ orthogonal matrix such that $O^{T}M^{T}MO$ is diagonal. Let $\bm{\widetilde{\Delta}}^{(k)}$ be the vector of $k\times k$ minors of $MO$. We claim that the norms $\lVert\bm{\widetilde{\Delta}}^{(k)}\rVert_{\infty}$ and $\lVert\bm{\Delta}^{(k)}\rVert_{\infty}$ are of comparable size.

Lemma 3.1 shows that $(MO)^{[k]}=M^{[k]}O^{[k]},$ and since $(O^{T})^{[k]}O^{[k]}=I^{[k]}$ and $(O^{T})^{[k]}_{\alpha\beta}=O^{[k]}_{\beta\alpha}$ we see that $O^{[k]}$ is orthogonal. Hence the maximum norm of the entries satisfies

[TABLE]

So in proving (3.2) we may assume that $M^{T}M$ is diagonal. The result follows. ∎

Part ii.

By permuting the rows and columns of $M,$ we may assume that

[TABLE]

Let $\bm{v}$ be in the span of the first $k$ basis vectors. If (3.3) holds for all such $\bm{v}$ then we have proved the lemma. Since $v_{i}=0$ for $i>k,$ one finds that

[TABLE]

where we have divided each matrix into three blocks, and $0_{p\times q}$ stands for a $p\times q$ block of zeroes. By (3.6) we have $\pm v_{1}\cdot\lVert\bm{\Delta}^{(k)}\rVert_{\infty}$ as the determinant of the left-hand side. Expanding the determinant of the right-hand side in the first column, we find that it is equal to

[TABLE]

Note the $(k-1)\times(k-1)$ determinant in which $i$ runs over $1,\dotsc,k$ with the value $\ell$ omitted, and $j$ runs over $2,\dotsc,k$.

By part i, this implies that $\Lambda_{k}v_{1}\ll_{m,n}\lVert M\bm{v}\rVert_{\infty},$ so provided that $\lVert\bm{v}\rVert_{\infty}=\lvert v_{1}\rvert,$ then (3.3) holds.

If we apply the same permutation both to the $v_{i}$ and to the first $k$ rows of $M,$ then both sides of our claim (3.3) and our assumption (3.6) remain the same. By applying such a permutation we may assume $\lVert\bm{v}\rVert_{\infty}=\lvert v_{1}\rvert,$ and so we have proved (3.3). ∎

Part iii.

Let $X$ be the span of the $\Lambda_{i}^{2}$-eigenvectors of $M^{T}M,$ where $i$ runs from $k$ up to $n$. As the matrix $M^{T}M$ is symmetric, we have $\bm{X}^{T}M^{T}M\bm{X}\ll_{n}\lVert\bm{X}\rVert_{\infty}^{2}\Lambda_{k}^{2}$ for all $\bm{X}\in X,$ and so

[TABLE]

for all $\bm{X}\in X.$ Therefore either this space $X$ satisfies (3.4), or the bound $\Lambda_{k}\gg_{m,n}C^{-1}$ holds and the existence of the space $V$ follows by part ii. ∎

4 Counting points in the sets $K_{k}(E_{1},\dotsc,E_{k+1})$

In this section our goal is to estimate the number of integer points in the sets $K_{k}(E_{1},\dotsc,E_{k+1})$ from Definition 2.1. We give the following result.

Lemma 4.1**.**

Let $c$ and $H_{c}(\bm{x})$ be as in Definition 1.1, let $\lambda_{c,i}(\bm{x}),$ $\bm{D}^{(c,i)}(\bm{x}),$ $J_{\bm{D}^{(c,i)}}(\bm{x})$ be as in Definition 1.2, and let $K_{k}(E_{1},\dotsc,E_{k+1})$ be as in Definition 2.1. Suppose that $B,C\geq 1,$ $\sigma\in\{0,\dotsc,n-1\},$ and $k\in\{0,\dotsc,n-\sigma-1\},$ and that $CB\geq E_{1}\geq E_{k+1}\geq 1$. Then at least one of the following holds:

1. (I)k

The set $K_{k}(E_{1},\dotsc,E_{k+1})$ may be covered by a collection of at most

[TABLE]

boxes in $\mathbb{R}^{n}$ of side $E_{k+1}$. Such a box contains $O_{n}(E_{k+1}^{n})$ integral points, so it follows that

[TABLE] 2. (II)k

There exist an integer $1\leq b\leq k,$ a point $\bm{x}^{(0)}\in K_{b}(E_{1},\dotsb,E_{b+1}),$ and a $(\sigma+b+1)$-dimensional linear subspace $X$ of $\mathbb{R}^{n}$ such that

[TABLE]

1. (III)

There is a $(\sigma+1)$-dimensional linear subspace $X$ of $\mathbb{R}^{n}$ such that

[TABLE]

with $\lVert H_{c}(\bm{X})\rVert_{\infty}$ as in §1.4.

We have subscripted the first two items to emphasize their dependence on $k$; note that item (III) has no such dependence.

In Corollary 5.1 below, we will use Lemma 4.1 to bound the quantities (2.1) and (2.2) from Corollary 2.1. Before proving the lemma, we give a comparison with step 2 in §1.4.

If there are many integer points $\bm{x}$ for which $\operatorname{rank}H_{c}(\bm{x})=b$ holds, then step 2 gives us a point $\bm{x}^{(0)}$ for which the matrix $J_{\bm{D}^{(c,b+1)}}(\bm{x}^{(0)})$ has a kernel of dimension $(\sigma+b+1)$ or more and $\operatorname{rank}H_{c}(\bm{x}^{(0)})=b$ holds.

If there are many integer points $\bm{x}$ for which $\bm{x}\in K_{k}(E_{1},\dotsc,E_{k+1}),$ then (4.1) is false and so either (II)k or (III) must hold. Of these, case (II)k gives us a point $\bm{x}^{(0)}$ such that $J_{\bm{D}^{(c,b+1)}}(\bm{x}^{(0)})$ is small on a $(\sigma+b+1)$-dimensional space. Moreover it states that $\bm{x}^{(0)}\in K_{b}(E_{1},\dotsb,E_{b+1})$ and that $E_{b+1}<C^{-1}E_{b},$ so that the $(b+1)$th eigenvalue of the matrix $H_{c}(\bm{x}^{(0)})$ is about $C$ times smaller than the $b$th eigenvalue. Thus (II)k gives us a point $\bm{x}^{(0)}$ for which in some sense $J_{\bm{D}^{(c,b+1)}}(\bm{x}^{(0)})$ is close to having a kernel of dimension at least $(\sigma+b+1)$ and $H_{c}(\bm{x}^{(0)})$ is close to having rank $b$.

The third case (III) is less directly comparable to step 2. We suggest that it could correspond to the case $b=0$ of step 2.

Proof of Lemma 4.1.

The proof is by induction on $k$. Let $c,$ $C,$ $B,$ and $\sigma$ be fixed. ∎

The case $k=0$.

Let $k=0,$ let $CB\geq E_{1}\geq 1$ and suppose that alternative (III) does not hold. We claim that alternative (I)0 holds, that is $K_{0}(E_{1})$ is covered by $O_{C,n}(B^{\sigma}/E_{1}^{\sigma})$ boxes of side $E_{1}$.

As (III)0 is false, applying Lemma 3.2ii to the matrix of the linear map $\bm{x}\mapsto H_{c}(\bm{x})$ shows that there is an $(n-\sigma)$-dimensional subspace $V$ of $\mathbb{R}^{n}$ with

[TABLE]

For each $\bm{z}\in\mathbb{R}^{n},$ let $A_{0}(\bm{z})$ be the box in $\mathbb{R}^{n}$ defined by

[TABLE]

Now $K_{0}(E_{1})$ is contained in the box $\lVert\bm{x}\rVert_{\infty}\leq B$. It follows that we can cover $K_{0}(E_{1})$ with a collection of $O_{C,n}(B^{\sigma}/E_{1}^{\sigma})$ boxes of the form $A_{0}(\bm{z}),$ each one of which is centred at a point $\bm{z}$ belonging to $K_{0}(E_{1})$. We will show below that for each $\bm{z}\in K_{0}(E_{1}),$ the intersection $A_{0}(\bm{z})\cap K_{0}(E_{1})$ is contained in a box of side $O_{C,n}(E_{1})$. It follows that $K_{0}(E_{1})$ is covered by $O_{C,n}(B^{\sigma}/E_{1}^{\sigma})$ boxes of side $E_{1},$ as claimed.

It remains to let $\bm{z}\in K_{0}(E_{1})$ and let $\bm{y}\in A_{0}(\bm{z})\cap K_{0}(E_{1}),$ and to deduce that $\lVert\bm{y}-\bm{z}\rVert_{\infty}\ll_{C,n}E_{1}$ must hold.

By definition of $K_{0}(E_{1})$ we have $\lvert\lambda_{c,1}(\bm{y})\rvert\leq E_{1}$ and $\lvert\lambda_{c,1}(\bm{z})\rvert\leq E_{1},$ and the bounds $\lVert H_{c}(\bm{y})\rVert_{\infty}\ll_{n}E_{1}$ and $\lVert H_{c}(\bm{z})\rVert_{\infty}\ll_{n}E_{1}$ follow by (1.10). So we have

[TABLE]

Let $\bm{u}\in V^{\perp}$ and let $\bm{v}\in V$ such that $\bm{y}=\bm{z}+\bm{u}+\bm{v}$ holds. Since $\bm{y}$ lies in $A_{0}(\bm{z}),$ we have $\lVert\bm{u}\rVert_{\infty}\leq E_{1},$ and with (4.5) this implies that

[TABLE]

By (4.4) it follows that $\lVert\bm{v}\rVert_{\infty}\ll_{n}CE_{1},$ and hence that $\lVert\bm{y}-\bm{z}\rVert_{\infty}\ll_{C,n}E_{1},$ as claimed. ∎

The inductive step.

Let $k\geq 1$ and let $CB\geq E_{1}\geq\dotsb E_{k+1}\geq 1$. We suppose that (II)k and (III) are both false, and claim that (I)k holds.

By induction, at least one of (I)k-1, (II)k-1, or (III) holds. Note that of these (III) is false by assumption, and (II)k-1 is false since it implies (II)k, and so (I)k-1 must hold.

Suppose for the time being that

[TABLE]

The contrary case is almost trivial and will be dealt with at the end of the proof. We claim that

[TABLE]

where $V$ runs over those $(n-\sigma-k)$-dimensional subspaces of $\mathbb{R}^{n}$ which are spanned by standard basis vectors, and we define

[TABLE]

We have assumed that $E_{k+1}<C^{-1}E_{k}$ and that the case $b=k$ of (II)k is false. So the case $b=k$ of (4.2) must be false for every $(\sigma+b+1)$-dimensional subspace $X$ of $\mathbb{R}^{n}$ and every $\bm{x}^{(0)}\in K_{k}(E_{1},\dotsc,E_{k+1})$.

That is, for any $\bm{x}^{(0)}\in K_{k}(E_{1},\dotsc,E_{k+1})$ and any $(\sigma+k+1)$-dimensional linear subspace $X$ of $\mathbb{R}^{n},$ there is some $\bm{X}\in X$ such that

[TABLE]

Applying Lemma 3.2ii with the choice $M=J_{\bm{D}^{(c,k+1)}}(\bm{x}^{(0)})$ shows that for each $\bm{x}^{(0)}\in K_{k}(E_{1},\dotsc,E_{k+1})$ there is an $(n-\sigma-k)$-dimensional subspace $V$ of $\mathbb{R}^{n},$ spanned by standard basis vectors, such that

[TABLE]

for all $\bm{v}\in V$. This proves (4.7), and so to prove (I)k it now suffices to show that for each $(n-\sigma-k)$-dimensional space $V,$ the set (4.8) is covered by a union of $O_{C,n}(B^{\sigma}(E_{1}\dotsm E_{k+1})E_{k+1}^{-\sigma-k-1})$ boxes of side $E_{k+1}$.

Let $\epsilon>0$ be a sufficiently small constant depending at most on $C$ and $n,$ and for each $\bm{z}\in\mathbb{R}^{n}$ set

[TABLE]

We assumed at the start of this inductive step that (I)k-1 holds, and so $K_{k-1}(E_{1},\dotsc,E_{k})$ is covered by a collection of $O_{C,n}(B^{\sigma}(E_{1}\dotsm E_{k})E_{k}^{-\sigma-k})$ boxes of side $E_{k}$. Subdivide each of these boxes into $O_{C,n}(E_{k}^{\sigma+k}/E_{k+1}^{\sigma+k})$ sub-boxes of the form $A_{k}(\bm{z})$. Since

[TABLE]

it follows that the set $K_{k}^{(C,V)}(E_{1},\dotsc,E_{k+1})$ may be covered by a collection of $O_{C,n}(B^{\sigma}(E_{1}\dotsm E_{k+1})E_{k+1}^{-\sigma-k-1})$ boxes of the form $A_{k}(\bm{z}),$ each of which is centred at a point $\bm{z}$ belonging to the set $K_{k}^{(C,V)}(E_{1},\dotsc,E_{k+1})$.

We will show below that for each such box $A_{k}(\bm{z}),$ the intersection $A_{k}(\bm{z})\cap K_{k}^{(C,V)}(E_{1},\dotsc,E_{k+1})$ is covered by a box of side $O_{C,n}(E_{k+1})$. It follows that each set (4.8) is covered by $O_{C,n}(B^{\sigma}(E_{1}\dotsm E_{k+1})E_{k+1}^{-\sigma-k-1})$ boxes of side $E_{k+1},$ and by the comments after (4.9) this proves the lemma.

We suppose that $\bm{z}\in K_{k}^{(C,V)}(E_{1},\dotsc,E_{k+1})$ and that $\bm{y}\in A_{k}(\bm{z})\cap K_{k}^{(C,V)}(E_{1},\dotsc,E_{k+1}),$ and we claim that $\lVert\bm{y}-\bm{z}\rVert_{\infty}\ll_{C,n}E_{k+1}$ holds. Let $\bm{u}\in V^{\perp}$ and let $\bm{v}\in V$ such that $\bm{y}=\bm{z}+\bm{u}+\bm{v},$ and note that since $\bm{y}\in A_{k}(\bm{z}),$ we have

[TABLE]

Now the $j$th partial derivatives of the $(k+1)\times(k+1)$ minors $\bm{D}^{(c,k+1)}(\bm{x})$ are linear combinations of the $(k+1-j)\times(k+1-j)$ minors $\bm{D}^{(c,k+1-j)}(\bm{x})$ with coefficients of size at most $O_{n}(1)$. So we have

[TABLE]

and Taylor expansion shows that

[TABLE]

It follows that

[TABLE]

Since $\bm{y},\bm{z}\in K_{k}(E_{1},\dotsc,E_{k+1}),$ Lemma 3.2 gives us the bounds

[TABLE]

and since $\bm{z}\in K_{k}^{(C,V)}(E_{1},\dotsc,E_{k+1})$ it follows from (4.8) that

[TABLE]

Substituting (4.13)-(4.14) into (4.12) yields

[TABLE]

Applying the bounds from (4.11) and the inequalities $E_{1}\geq\dotsb\geq E_{k+1},$ we deduce that

[TABLE]

Since $\epsilon$ is assumed to be small in terms of $C$ and $n,$ it follows that $\lVert\bm{v}\rVert_{\infty}\ll_{n}CE_{k+1}$ holds and hence that $\lVert\bm{y}-\bm{z}\rVert_{\infty}\ll_{C,n}E_{k+1}$ holds. By the comments after (4.10), this proves the lemma.

It remains to consider the case when (4.6) is false and so $E_{k+1}\geq C^{-1}E_{k}$ holds. At the start of the inductive step we supposed that (I)k-1 holds, so the set $K_{k-1}(E_{1},\dotsc,E_{k})$ may be covered by $O_{C,n}(B^{\sigma}(E_{1}\dotsm E_{k})E_{k}^{-\sigma-k})$ boxes of side $E_{k}$. We have

[TABLE]

and so the set $K_{k}(E_{1},\dotsc,E_{k+1})$ is also covered by this collection of boxes. Since $E_{k+1}\geq C^{-1}E_{k}$ holds, we can divide each of these boxes into $O_{C,n}(1)$ boxes of side $E_{k+1}$. This proves (I)k. ∎

5 Small values of a trilinear form

Part 3 of Davenport’s argument from §1.4 starts from a point $\bm{x}$ for which the matrices $H_{c}(\bm{x})$ and $J_{\bm{D}^{(c,b+1)}}(\bm{x})$ have prescribed ranks, and finds linear spaces $X,Y$ such that for all $\bm{X}\in X$ and $\bm{Y},\bm{Y}^{\prime}\in Y$ the equation $\bm{Y}^{T}H_{c}(\bm{X})\bm{Y}^{\prime}=0$ holds. Our analogue is the following pair of results, which give linear spaces on which the trilinear form $\bm{Y}^{T}H_{c}(\bm{X})\bm{Y}^{\prime}$ is small.

Lemma 5.1**.**

Let $c(\bm{x})$ be as in Definition 1.1, and let $\lambda_{c,i}(\bm{x})$ and $J_{\bm{D}^{(c,i)}}(\bm{x})$ be as in Definition 1.2. Suppose that $b\in\{1,\dotsc,n-1\}$ and that $\bm{x}^{(0)}\in\mathbb{R}^{n}$. Then, provided $\bm{D}^{(c,b)}(\bm{x}^{(0)})$ is nonzero, there exists an $(n-b)$-dimensional linear subspace $Y$ of $\mathbb{R}^{n}$ such that for all $\bm{Y},\bm{Y}^{\prime}\in Y$ and all $\bm{t}\in\mathbb{R}^{n}$ we have

[TABLE]

By setting $b=\operatorname{rank}H_{c}(\bm{x}^{(0)})$ and $X=\ker J_{\bm{D}^{(c,b+1)}}(\bm{x}^{(0)}),$ one may recover Davenport’s result. We prove Lemma 5.1 at the end of this section, after deducing

Corollary 5.1**.**

Let $c,$ $H_{c}(\bm{x})$ and $N^{\operatorname{aux}}_{c}(B)$ be as in Definition 1.1. For any $B,C\geq 1$ and any $\sigma\in\{0,\dotsc,n-1\},$ one of the following alternatives holds. Either

[TABLE]

or there exist positive-dimensional linear subspaces $X$ and $Y$ of $\mathbb{R}^{n}$ for which $\dim X+\dim Y=n+\sigma+1,$ such that

[TABLE]

Proof.

We saw in Lemma 4.1 that for any $k\in\{0,\dotsc,n-\sigma-1\}$ and any $E_{1},\dotsc,E_{k+1}\in\mathbb{R}$ satisfying

[TABLE]

one of (I)k, (II)k, or (III) must hold. Suppose first that in every case alternative (I)k holds. By (4.1), we then have

[TABLE]

for every $k\in\{0,\dotsc,n-\sigma-1\}$ and every $CB\geq E_{1}\geq\dotsb\geq E_{k+1}\geq 1$. Now Corollary 2.1 shows that either

[TABLE]

or

[TABLE]

where $k\in\{1,\dotsc,n-1\}$ and the inequalities $B\gg_{n}2^{e_{1}}\geq\dotsb\geq 2^{e_{k+1}}\geq 1$ hold, or

[TABLE]

where the inequalities $B\gg_{n}2^{e_{1}}\geq\dotsb\geq 2^{e_{n}}\geq 1$ hold. We may assume that $C$ is sufficiently large in terms of $n,$ so that $CB\geq 2^{e_{1}}$ holds in (5.6)-(5.7). Substituting the bound (5.4) into each of (5.5)-(5.7) proves the conclusion (5.2).

Suppose next that alternative (III) holds in Lemma 4.1. In this case we let $Y=\mathbb{R}^{n},$ and the conclusion (5.3) follows from (4.3).

It remains to treat the case when there exist $k\in\{0,\dotsc,n-\sigma-1\}$ and $CB\geq E_{1}\geq\dotsb\geq E_{k+1}\geq 1$ such that alternative (II)k holds in Lemma 4.1. This means that there exist an integer $1\leq b\leq k,$ a point $\bm{x}^{(0)}\in K_{b}(E_{1},\dotsb,E_{b+1}),$ and a $(\sigma+b+1)$-dimensional linear subspace $X$ of $\mathbb{R}^{n}$ such that

[TABLE]

Since $\bm{x}^{(0)}\in K_{k}(E_{1},\dotsc,E_{k+1}),$ the inequalities $\frac{1}{2}E_{i}<\lambda_{c,i}(\bm{x}^{(0)})\leq E_{i}$ hold. Therefore (5.8) implies

[TABLE]

Note that (5.10) implies that $\lambda_{c,b}(\bm{x}^{(0)})$ is nonzero, and so $\bm{D}^{(c,b)}(\bm{x}^{(0)})$ is nonzero, by Lemma 3.2. Hence we may apply Lemma 5.1 to find an $(n-b)$-dimensional linear space $Y$ such that for all $\bm{Y},\bm{Y}^{\prime}\in Y$ and all $\bm{t}\in\mathbb{R}^{n}$ the bound (5.1) holds. The conclusion (5.3) follows on taking $\bm{t}=\bm{X}$ in (5.1) of Lemma 5.1 and substituting in the bounds (5.9) and (5.10). ∎

Proof of Lemma 5.1.

We imitate the proof of Lemma 3 in Davenport [12], which begins by considering the following easy “warm-up” problem. Suppose we were to look for $n-b$ linearly independent vectors $\bm{y}$ at which $H_{c}(\bm{x}^{(0)})\bm{y}$ vanishes. One approach would be as follows. One can construct matrices $L^{(i)},$ $M^{(i)}$ for $i=1,\dotsc,n-b,$ with entries in $\{0,\pm 1\},$ such that the vectors

[TABLE]

That is, the components of $\bm{y}^{(i)}(\bm{x})$ are polynomials of the form $\pm\Delta_{j}^{(b)}(\bm{x}),$ and the components of $H_{c}(\bm{x})\bm{y}^{(i)}(\bm{x})$ are polynomials of the form $\pm\Delta_{j}^{(b+1)}(\bm{x})$.

If $H_{c}(\bm{x}^{(0)})\bm{y}=\bm{0}$ had exactly $n-b$ linearly independent solutions $\bm{y},$ we would have $\bm{D}^{(c,b+1)}(\bm{x}^{(0)})=\bm{0},$ while $\bm{D}^{(c,b)}(\bm{x}^{(0)})$ would be nonzero. We would then have $n-b$ solutions $\bm{Y}^{(k)}$ defined by

[TABLE]

and if we chose our matrices $L^{(i)},M^{(i)}$ appropriately these would be linearly independent.

We now return to the proof of the lemma. Assume for the time being that $L^{(i)},$ $M^{(i)},$ $\bm{y}^{(i)}(\bm{x})$ and $\bm{Y}^{(i)}$ satisfying (5.11)-(5.13) are given, and let $\bm{x}^{(0)}$ be as in the lemma. Let $\bm{t}\in\mathbb{R}^{n}$. Let $\partial_{\bm{t}}$ be the directional derivative along $\bm{t}$ defined by $\sum t_{i}\frac{\partial}{\partial x_{i}},$ and apply $\partial_{\bm{t}}$ to both sides of (5.12). This shows that

[TABLE]

Since $\partial_{\bm{t}}$ is a linear operator and we have

[TABLE]

it follows from (5.14) that

[TABLE]

Now Lemma 3.2 shows that

[TABLE]

and substituting these bounds into (5.15) gives

[TABLE]

where the $\bm{Y}^{(k)}$ are as in (5.13).

The idea is now to let $Y$ be the span of the $\bm{Y}^{(k)}$ and deduce (5.1) from (5.16). Since we are looking for an $(n-b)$-dimensional space $Y$ we will need $\bm{Y}^{(1)},\dotsc,\bm{Y}^{(n-b)}$ to be linearly independent. In order to prove (5.1) we require the following slightly stronger statement. We claim there are $L^{(i)},$ $M^{(i)},$ $\bm{y}^{(i)}(\bm{x})$ and $\bm{Y}^{(i)}$ satisfying (5.11)-(5.13), such that the linear combination defined by $\bm{Y}=\sum_{i=1}^{n-b}\gamma_{i}\bm{Y}^{(i)}$ satisfies $\lVert\bm{\gamma}\rVert_{\infty}\ll_{n}\lVert\bm{Y}\rVert_{\infty}$ for every vector $\bm{\gamma}$ in real $(n-b)$-space. The lemma then follows, with $Y$ being the span of the $\bm{Y}^{(i)},$ on expressing $\bm{Y},\bm{Y}^{\prime}$ as linear combinations of the $\bm{Y}^{(i)}$ and applying (5.16).

For the remainder of the proof we will assume for simplicity that the $b\times b$ minor of $H_{c}(\bm{x}^{(0)})$ with largest absolute value is the minor in the lower right-hand corner, that is, we will assume that

[TABLE]

In general (5.17) holds after permuting the rows and columns of the matrix $H_{c}(\bm{x})$ and one can then apply the same permutations throughout the rest of our construction of $\bm{Y}^{(i)},$ every time the matrix $H_{c}(\bm{x})$ appears.

Define $\bm{y}^{(1)}(\bm{x}),\dotsc,\bm{y}^{(n-b)}(\bm{x})$ by

[TABLE]

where $(\ell=i,n-b+1,\dotsc,n;\;\ell\neq j)$ means that $\ell$ first takes the value $i$ and then runs over the numbers $n-b+1,\dotsc,n$ with $j$ omitted. Now this is of the form (5.11), and one can check that

[TABLE]

which is of the form (5.12). Define a matrix $Q$ by

[TABLE]

or equivalently by

[TABLE]

so that the entries $Q_{ij}$ have absolute value at most 1. Then one sees from (5.17) that

[TABLE]

where $\widetilde{Q}$ is some $(n-b)\times(n-b)$ matrix. In particular $\det Q=1,$ and so the entries of ${Q^{-1}}$ are bounded in terms of $n$. It follows that if $\bm{Y}=\sum_{i=1}^{n-b}\gamma_{i}\bm{Y}^{(i)}$ then $\gamma_{i}=(Q^{-1}\bm{Y})_{i}\ll_{n}\lVert\bm{Y}\rVert_{\infty},$ as claimed. ∎

6 Constructing singular points on $V(c)$

Corollary 5.1 shows that either $N^{\operatorname{aux}}_{c}(B)$ is small, or there are spaces $X,Y$ of large dimension on which $\bm{Y}^{T}H_{c}(\bm{X})\bm{Y}^{\prime}$ is small. To prove Proposition 1.1 we show that the second alternative implies that $V(c)$ is singular. This is our analogue of Davenport’s step 4, as described in §1.4.

Proof of Proposition 1.1.

Suppose for a contradiction that the result is false. Then for every $N\in\mathbb{N}$ there is $c_{N}\in{\mathcal{K}}$ with

[TABLE]

By Corollary 5.1, this implies that there are linear subspaces $X_{N},Y_{N}$ of $\mathbb{R}^{n}$ such that

[TABLE]

holds and for all $\bm{X}\in X_{N}$ and $\bm{Y},\bm{Y}^{\prime}\in Y_{N},$ we have

[TABLE]

If we multiply $c_{N}$ by a constant then the matrix $H_{c_{N}}(\bm{x})$ does not change. So we may assume that for each $N$ the equality $\lVert c_{N}\rVert_{\infty}=1$ holds. After passing to a subsequence we have $c_{N}\to c$ as $N\to\infty,$ and it follows that there are subspaces $X,Y$ of $\mathbb{R}^{n}$ such that $\dim X+\dim Y=n+\sigma_{\mathcal{K}}+1$ and

[TABLE]

Let $b\in\{0,\dotsc,n-\sigma-1\}$ such that

[TABLE]

Let $\bm{x}^{(1)},\dotsc,\bm{x}^{(n)}$ be a basis of $\mathbb{R}^{n}$ such that $\bm{x}^{(b+1)},\dotsc,\bm{x}^{(n)}$ is a basis of $X$.

Let $[Y]$ be the projective linear space in $\mathbb{P}_{\mathbb{R}}^{n-1}$ associated to $Y$. Take homogeneous co-ordinates $\bm{y}$ on $[Y],$ so that $\bm{y}$ takes values in $Y$.

Let $W$ be the projective variety cut out in $[Y]$ by the $b$ equations

[TABLE]

so that

[TABLE]

We claim that $W$ is contained in the singular locus of the projective hypersurface $V(c)$. It follows that $\dim\operatorname{Sing}V(c)\geq\sigma_{\mathcal{K}},$ which is a contradiction, by (1.6).

Now (6.1) implies that for every $\bm{y}\in Y$ we have

[TABLE]

So if we let $\bm{y}\in Y$ such that (6.2) holds, then we have

[TABLE]

This implies that $\bm{\nabla}_{\bm{y}}c(\bm{y})=\bm{0}$ holds, by the definition (1.4). It follows that every point of $W$ is contained in $\operatorname{Sing}V(c),$ as claimed. ∎

7 Acknowledgements

This work is based on a DPhil thesis submitted to Oxford University, and was supported by EPSRC grants EP/J500495/1 and EP/M507970/1. I would like to thank my DPhil supervisor, Roger Heath-Brown. I am grateful to Rainer Dietmann for helpful conversations.