A Note on Thue Inequalities with Few Coefficients

Abstract

Let $F(X,Y)=\sum \limits _{i=0}^sa_iX^{r_i}Y^{r-r_i}\in \mathbb {Z}[X,Y]$ be a form of degree $r\ge 3$, irreducible over $\mathbb {Q}$, and having at most $s+1$ nonzero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality

$$ |F(X,Y)|\le h $$

is $\ll s^2h^{2/r}(1+\log h^{1/r})$. They conjectured that $s^2$ may be replaced by s. In this note we show some instances when $s^2$ may be improved.

This paper is dedicated to Krishna Alladi on the occasion of his 60th birthday

1 Introduction

Let F(X, Y) be a form of degree $r\ge 3$ with integer coefficients, irreducible over $\mathbb {Q}$, and having at most $s+1$ nonzero coefficients. Write

$$\begin{aligned} F(X,Y)=\sum \limits _{i=0}^sa_iX^{r_i}Y^{r-r_i} \end{aligned}$$

(1.1)

with $0=r_0<r_1<\ldots <r_s=r$. Let D, H, and M denote the discriminant, naive height, and Mahler height of F(X, 1), respectively. For $h\ge 1$, consider the Thue inequality

$$\begin{aligned} |F(X,Y)|\le h. \end{aligned}$$

(1.2)

Let $N_F(h)$ denote the number of integer solutions (x, y) of (1.2). Bombieri modified a conjecture of Siegel and asked if $N_F(h)$ could be bounded by a function depending only on s and h. (See Mueller and Schmidt [8, p. 208]). Toward this, Schmidt [10] proved that

$$\begin{aligned} N_F(h)\ll \sqrt{rs}\ h^{2/r}(1+\log h^{1/r}). \end{aligned}$$

(1.3)

Throughout this note, the constants implied by $\ll$ are absolute. The modified Siegel’s conjecture was shown to be true in the case $s=1$ by Hyyrö [3], Evertse [1], and Mueller [6]. The case $s\ge 2$ was considered by Mueller and Schmidt in [7] and [8]. They proved that

$$\begin{aligned} N_F(h)\ll s^2C(r,h) \end{aligned}$$

(1.4)

where $C(r,h)=h^{2/r}(1+\log h^{1/r})$. From a result of Mahler [5], it is known that the factor $h^{2/r}$ in C(r, h) is unavoidable while the logarithmic factor was improved by Thunder when h is large, see [11] and [12].

When s is as large as r, (1.4) is weaker than (1.3). It was conjectured in [8] that it may possible to replace the factor $s^2$ above by s. In [9], some results were given where the factor $s^2$ was improved. For instance, the following results were proven.

• (i) We always have $|r_i-r_w| \ge |i-w| \ge 1\ \mathrm{for}\ i \ne w.$ Suppose $|r_i-r_w|\ge c_1|i-w|$ with $c_1\ge 1,$ an absolute constant. Then,

  $$ N_F(h)\ll s^{1+\frac{1}{c_1}}\ C(r,h). $$

  Thus, the exponent of s is $<2$ whenever $c_1>1.$

• (ii) Suppose $|r_i-r_w|\ge \frac{1}{3}|i-w|\log |i-w|$. Then,

  $$ N_F(h)\ll s \log ^3 s\ C(r,h). $$

In another direction, it was shown that if the coefficients of F(X, Y) satisfy

$$\begin{aligned} \left| \frac{a_i}{a_0}\right| ^{1/r_i}\le \ \left| \frac{a_s}{a_0} \right| ^{1/r_s}\text { for } i=1,\ldots ,s-1 \end{aligned}$$

(1.5)

and $r\ge \max (4s,s\log ^3s)$, then

$$\begin{aligned} N_F(h)\ll s(\log s) \ C(r,h). \end{aligned}$$

(1.6)

If $r<\max (4s,s\log ^3 s)$, then by (1.3), we have

$$ N_F(h)\ll s(\log s)^{3/2} \ C(r,h). $$

Thus, under the condition (1.5), we have

$$N_F(h) \ll s (\log s)^{3/2}\ C(r,h).$$

In particular, the above estimate holds whenever $|a_0|=|a_s|=H$ where H is the maximum of the absolute values of the coefficients of F. Hence, the estimate holds for forms F with coefficients $\pm 1.$

Let $q_1$ be the smallest integer, $0\le q_1\le s$, with $|a_{q_1}|=H$ and let $q_2$ be the largest integer, $0\le q_2\le s$, with $|a_{q_2}|=H$. The condition (1.5) implies that

$$ (q_1,q_2)\in \{(0,0), (0,s), (s,s)\}. $$

In this note, we shall consider a few more cases of $(q_1,q_2)$. Throughout, we use the following assumptions A.

• A:

  • (a) $r\ge \max (4s,s\log ^3s)$.

  • (b) $|a_i|^{r_{q_1}} \le |a_{q_1}|^{r_i}|a_0|^{r_{q_1}-r_i}$ for $0\le i \le q_1$.

  • (c) $|a_i|^{r-r_{q_2}} \le |a_{q_2}|^{r-r_i}|a_s|^{r_i-r_{q_2}}$ for $q_2\le i \le s$.

Note that A(b) holds trivially if $q_1=0$ and A(c) holds trivially if $q_2=s$. We prove the following result.

Theorem 1.1

Suppose that the assumption A holds. Then, (1.6) is valid in the following three cases:

• (i) $q_1=0$, $0<q_2<s$ and $H\le |a_s|^{\frac{r}{\max (s,r_{q_2})}}$;

• (ii) $q_2=s$, $0<q_1<q_2$ and $H\le |a_0|^{\frac{r}{\max (s,r-r_{q_1})}}$;

• (iii) $q_1\ne 0$, $q_2\ne s$ and $H\le \min \left( |a_0|^{\frac{r-s}{r}}|a_s|^{\frac{r_{q_1}}{r}},|a_0|^{\frac{r-r_{q_2}}{r}}|a_s|^{\frac{r-s}{r}}\right)$.

Remark.

• (a) When $r<\max (4s,s \log ^3s)$, we use (1.3) to obtain

  $$ N_F(h)\ll s(\log s)^{3/2}\ C(r,h). $$

  Therefore, under the conditions of the theorem, we have

  $$N_F(h) \ll s(\log s)^{3/2}\ C(r,h).$$

• (b) We may assume that s is large, as otherwise inequality (1.4) is sufficient.

2 Preliminaries

Let F(X, Y) be given by (1.1). Let $X_1$, $X_2>0$. Divide the solutions (x, y) of (1.2) into three sets as

$$\begin{aligned} \max (|x|,|y|)>X_1;&\max (|x|,|y|)\le X_1\, \text {and} \min (|x|,|y|)\ge X_2; \\ \nonumber&\,\,\, \min (|x|,|y|)< X_2. \end{aligned}$$

Denote the number of primitive solutions in these sets by $P_{lar}(X_1)$, $P_{med}(X_1,X_2)$, and $P_{sma}(X_2)$, respectively. If $X_2>X_1,$ put $P_{med}(X_1,X_2)=0.$ Let P(h) be the number of primitive solutions of (1.2). Thus,

$$P(h)=P_{lar}(X_1) +P_{med}(X_1,X_2)+P_{sma}(X_2).$$

We can bound $N_F(h)$ by finding an upper estimate for P(h). Choose numbers a, b with $0<a<b<1.$ Define

$$t= \sqrt{2/(r+a^2)}, \lambda =2/((1-b)t),$$

$$ A=\frac{1}{a^2}\left( \log M+\frac{r}{2}\right) .$$

Further, we put

$$ B=\frac{2^{r}r^{r/2}M^rh}{\sqrt{|D|}}, R=e^{800\log ^3r}, $$

$$ Y_E=(2B\sqrt{|D|})^{1/(r-\lambda )}(4e^A)^{\lambda /(r-\lambda )}, Y_W=R^{1/(r-\lambda )}Y_E $$

$$ Y_S=(4^r(rs)^{2s}R^sh)^{\frac{1}{r-2s}}. $$

We take

$$\begin{aligned} X_1=Y_W\ \mathrm{and}\ X_2=Y_S. \end{aligned}$$

Mueller and Schmidt [8] had shown that

$$\begin{aligned} P_{lar}(Y_W)\ll s. \end{aligned}$$

(2.1)

To estimate the number of small solutions, we use the following lemma from [8].

Lemma 2.1

[8, Lemma 18]

Let F(X, Y) be given by (1.1) and let $r\ge 4s$. Then for any $Y\ge 1$, we have

$$ P_{sma}(Y) \ll (rs^2)^{2s/r} h^{2/r} + s\ Y. $$

The next lemma is a consequence of the above lemma.

Lemma 2.2

Let F(X, Y) be given by (1.1). Then,

$$ P_{sma}(Y_S)\ll s\ h^{2/r} { whenever}\; r\ge s\log ^3s. $$

For dealing with the medium solutions, we use the Archimedean Newton polygon of the polynomial F(X, 1). This is the lower boundary of the convex hull of the points $P_i=(r_i,-\log |a_i|), 0\le i \le s.$ Let $L_{i,j}$ denote the line joining $P_i$ to $P_j$ with slope $\sigma (i,j),$ say. Further, let

$$\hat{F}(X,1)= a_s+a_{s-1}X^{r_s-r_{s-1}}+\cdots +a_1X^{r_s-r_1}+a_0X^{r_s}$$

be the reciprocal polynomial of F(X, 1). Let $Q_i=(r_s-r_i,-\log |a_{i}|)$ and let $L'_{i,j}$ denote the line joining $Q_i$ to $Q_j$ with slope $\sigma '(i,j)$.

Lemma 2.3

Suppose that the coefficients of F(X, Y) satisfy the assumptions A(b) and A(c). Then, the edges of the Archimedean Newton polygon of F(X, 1) are $L_{0,q_1},L_{q_1,q_2}$, and $L_{q_2,s}.$ Further, $\sigma (q_1,q_2)=0$ and every root $\alpha$ of F(x, 1) satisfies

$$\begin{aligned} \frac{1}{2} e^{\sigma (0,q_1)}<|\alpha |<2 e^{\sigma (q_2,s)}. \end{aligned}$$

(2.2)

Every root $\beta$ of $\hat{F}(X,1)$ satisfies

$$ \frac{1}{2} e^{-\sigma (q_2,s)}<|\beta |<2 e^{-\sigma (0,q_1)}. $$

Proof

Put

$$ \sigma _1=\sigma (0,q_1)\ \text { and }\ \sigma _2=\sigma (q_2,s). $$

By A(b), we have

$$\sigma (0,i)\ge \sigma _1 \text { for } 0 \le i \le q_1.$$

By A(c), we have

$$\sigma (q_2,i)\ge \sigma _2 \text { for } q_2 \le i \le s.$$

Hence, the Archimedean Newton polygon consists of $L_{0,q_1}$ and $L_{q_2,s}$ as the left most edge and right most edge, respectively. Since the height of the polynomial is attained at $a_{q_1}$ and at $a_{q_2}$, we see that $\sigma (q_1,q_2)=0$ and $\sigma (q_1,i)\ge 0$ for $q_1 \le i \le q_2.$ Thus, $L_{q_1,q_2}$ is the third edge. Now we prove (2.2). By the convexity of the Newton polygon, we have

$$\begin{aligned} \sigma _2\ge \sigma (i,s) \text { for } 0 \le i \le s. \end{aligned}$$

(2.3)

Let $z=e^{\sigma _2 w}$ with $|w|\ge 2.$ Then,

$$|F(z,1)|\ge |a_s|e^{\sigma _2r_s}\left( |w|^{r_s}- \frac{|a_{s-1}|}{|a_s|}e^{\sigma _2(r_{s-1}-r_s)}|w|^{r_{s-1}}- \cdots -\frac{|a_{0}|}{|a_s|}e^{-\sigma _2r_s}\right) .$$

By (2.3), we have

$$\frac{|a_i|}{|a_s|}e^{\sigma _2(r_i-r_s)}\le 1.$$

Hence,

$$|F(z,1)| \ge |w|^{r_s}-|w|^{r_{s-1}}-\cdots -1>0$$

since $|w|\ge 2.$ Thus, every root $\alpha$ of F(X, 1) has $|\alpha |<2 e^{\sigma _2}.$

To get the lower bound, we use the reciprocal polynomial $\hat{F}(X,1).$ The Archimedean Newton polygon of this polynomial has edges $L'_{s,q_2}$, $L'_{q_2,q_1}$, and $L'_{q_1,0}.$ Arguing as above, we find that every root $\beta$ which is the inverse of some root $\alpha$ of F(X, 1) satisfies $|\beta |\le 2 e^{\sigma '(q_1,0)}$, where $\sigma '(q_1,0)$ is the slope of $L'_{q_1,0}$. Hence,

$$|\alpha | \ge \frac{1}{2} e^{-\sigma '(q_1,0)}.$$

Now the result follows on noticing that $\sigma '(q_1,0)=-\sigma _1$.               $\square$

The Archimedean Newton polygons when conditions (i), (ii), or (iii) of Theorem 1.1 hold, are shown in Figure 1.

Fig. 1

Archimedean Newton polygon.

Another tool needed to estimate the medium solutions is the Diophantine approximation property. Let S be the set of roots $\alpha _1,\cdots , \alpha _r$ of $f(z)=F(z,1)$ and $S^*$ the set of roots of F(1, z). Then, $S^*=\{\alpha _1^{-1},\cdots , \alpha _r^{-1}\}.$ Let (x, y) be a solution of (1.2) with $y\ne 0$. Define

$$d\left( S,\ \frac{x}{y}\right) =\min _{1 \le i \le r}\left| \alpha _i-\frac{x}{y}\right| .$$

It was shown in [8, Lemma 7], that there exists $S_1 \subseteq S$ with $|S_1| \ll s$ such that

$$\begin{aligned} d\left( S_1,\ \frac{x}{y}\right) \le R\ d\left( S,\ \frac{x}{y}\right) . \end{aligned}$$

(2.4)

Suppose that $d(S,x/y)=|\alpha -x/y|$ for some $\alpha \in S$. If $f^{(u)}(\alpha ) \ne 0$ for some u with $1 \le u \le r$, then by [8, Lemma 10], we have

$$\begin{aligned} \left| \alpha -\frac{x}{y}\right| \le \frac{r}{2} \left( \frac{2^rh}{|f^{(u)}(\alpha )||y|^r}\right) ^{1/u}. \end{aligned}$$

(2.5)

Let e, h be two nonnegative integers. Let $(e)_h$ be the Pochhammer symbol defined as

$$ (e)_h={\left\{ \begin{array}{ll} 0 &{}\text{ if } e=0\\ 1 &{}\text{ if } h=0\\ e(e-1)\cdots (e-h+1) &{}\text{ otherwise. } \end{array}\right. } $$

Using the explanations given in [8, p. 223–231], we can obtain

$$\begin{aligned} \sum _{u=1}^s E_u^{(s)} \alpha ^u f^{(u)}(\alpha )= a_s \alpha ^{r_s}\prod _{0\le i<j\le s}(r_i-r_j) \end{aligned}$$

(2.6)

where

$$E_u^{(s)}=(-1)^{s+u}\det \begin{pmatrix} 1 &{} \cdots &{} 1\\ (r_0)_1&{} \cdots &{} (r_{s-1})_1\\ \vdots &{} \vdots &{} \vdots \\ (r_0)_{u-1}&{} \cdots &{} (r_{s-1})_{u-1}\\ (r_0)_{u+1}&{} \cdots &{} (r_{s-1})_{u+1}\\ \vdots &{} \vdots &{} \vdots \\ (r_0)_{s}&{} \cdots &{} (r_{s-1})_{s} \end{pmatrix}. $$

From [8, Eqns (6.12) & (6.13)], we get that

$$|E_u^{(s)}| \le 2^s(s^2r)^{s-1}\prod _{0\le i<j\le s}(r_j-r_i).$$

We also refer to [9] for more details. Using the above estimate for $|E_u^{(s)}|$ in (2.6), we find that there exists u with $1 \le u \le s$ such that

$$|f^{(u)}(\alpha )| \ge |a_s||\alpha |^{r-u} 2^{-s}(s^2r)^{-(s-1)}s^{-1}.$$

The following lemma is now immediate from (2.4) and (2.5).

Lemma 2.4

There exists a set $S_1 \subseteq S$ with $|S_1| \ll s,$ such that for some $\alpha \in S_1,$ we have

$$\left| \alpha -\frac{x}{y}\right| \le \frac{rR}{2}\left( \frac{s(rs^2)^{s-1}2^{r+s}h}{|y|^r|a_s| |\alpha |^{r-u}}\right) ^{1/u}.$$

A similar inequality holds with (x, y) replaced by (y, x) for some set $S_2\subseteq S^*$ of roots with $|S_2|\le s.$

The following is a lemma on counting the number of elements in a set satisfying some gap conditions. (See [9, Lemma 2.1(i)]).

Lemma 2.5

Let $n \ge 2$ and let $U=\{u_1, \cdots , u_n\}$ be a set together with a map $T: U \rightarrow \mathbb R^*$ such that

$$A_1 \le T(u_1) \le T(u_2)\le \cdots \le T(u_n)$$

and

$$T(u_i) \ge \beta T(u_{i-1})^{\gamma }\ for \ 2\le i \le n\ with \ \beta >0, \gamma \ge 2.$$

Let

$$\kappa = {\left\{ \begin{array}{ll} 2\ if \ \beta >1\\ 1\ if \ \beta \le 1. \end{array}\right. }$$

Suppose that $T(u_n)\le B_1$ and $A_1\beta ^{1/(\kappa (\gamma -1))}>1.$ Then,

$$n \le 1+\frac{1}{\log \gamma } \log \left( \frac{\log B_1}{\log A_1+(\log \beta )/(\kappa (\gamma -1))}\right) .$$

Proof

By induction, we get

$$\begin{aligned} \nonumber T(u_{n})\ge & {} \beta ^{1+\gamma +\cdots +\gamma ^{n-2}} T(u_1)^{\gamma ^{n-1}}\\\ge & {} (\beta ^{1/(\kappa (\gamma -1))}T(u_1))^{\gamma ^{n-1}}. \end{aligned}$$

(2.7)

Since $T(u_n)\le B_1$, from (2.7), we get

$$ (\beta ^{1/(\kappa (\gamma -1))}T(u_1))^{\gamma ^{n-1}}\le B_1. $$

Taking logarithms twice, we get the assertion of the lemma.         $\square$

3 Proof of Theorem 1.1

By (2.1) and Lemma 2.1, it is enough to estimate $P_{med}(Y_W,Y_S).$ We shall consider $S_1$ from Lemma 2.4. The argument for $S_2$ is similar. We claim that

$$ |a_s \alpha ^{r-u}|\ge H^{u/r}2^{-(r-u)}. $$

We prove this when condition (ii) of the theorem holds. The other cases are similar. By Lemma 2.3,

$$\begin{aligned} |a_s \alpha ^{r-u}|> & {} |a_s|e^{(r-u)\sigma _1}2^{-(r-u)}\\= & {} |a_{s}|e^{(r-u)\frac{(-\log H+\log |a_0|)}{r_{q_1}}}2^{-(r-u)}. \end{aligned}$$

Thus, our claim is true if

$$ |a_s||a_0|^{\frac{r-u}{r_{q_1}}}\ge H^{\frac{r-u}{r_{q_1}}+\frac{u}{r}}. $$

Since $q_2=s$, we have $|a_s|=H$. Therefore, the above inequality holds if

$$ |a_0|\ge H^{\frac{r-r_{q_1}}{r}}, $$

which is true by our assumption. Hence, the claim follows. Thus, by Lemma 2.4, for $y \ge Y_S$,

$$\begin{aligned} \left| \alpha -\frac{x}{y}\right| < \frac{rR}{2H^{1/r}}\left( \frac{s2^{2r}(rs^2)^{s-1}h}{y^r}\right) ^{1/s}. \end{aligned}$$

(3.1)

Let $U=\{(x_1,y_1),\cdots , (x_\nu ,y_\nu )\}$ be the set of all solutions of (3.1) with gcd$(x_i,y_i)=1$ and

$$ Y_S \le y_1 \le \cdots \le y_\nu \le Y_W.$$

Suppose $\nu \ge 2.$ Then,

$$\frac{1}{y_iy_{i+1}}\le \left| \frac{x_i}{y_i}- \frac{x_{i+1}}{y_{i+1}}\right| \le \left| \alpha -\frac{x_{i}}{y_{i}}\right| + \left| \alpha -\frac{x_{i+1}}{y_{i+1}}\right| $$

$$\le \frac{K}{2y_i^{r/s}}+\frac{K}{2y_{i+1}^{r/s}}\le \frac{K}{y_i^{r/s}},$$

where

$$ K=R(rs)^24^{r/s}h^{1/s}H^{-1/r}. $$

Thus, we have

$$y_{i+1} \ge K^{-1}y_i^{r/s-1}.$$

We apply Lemma 2.5 with $T((x_i,y_i))=y_i, \beta = 1/K, \gamma = \frac{r-s}{s}, A_1=Y_S$ and $B_1=Y_W.$ Note that $\gamma =r/s-1 \ge \max (3,\log ^3s-1).$ Also, $R \ge 4(rs)^4.$ Further, $\log Y_W \ll \sqrt{r} +\log H+\log h^{1/r}.$ Hence, by Lemma 2.5, we get

$$\begin{aligned} \nu\ll & {} 1+\frac{1}{\log \gamma }\log \left( \frac{\log Y_W}{\log Y_S+\frac{\log \beta }{\kappa (\gamma -1)}}\right) \\\ll & {} \frac{1}{\log \gamma }\log \left( \frac{2(r-2s) (\sqrt{r}+\log H +\log h^{1/r})}{\log H}\right) \\\ll & {} \frac{\log r+\log (1+\log h^{1/r})}{\log \gamma }. \end{aligned}$$

Suppose $r \ll s^3.$ Then,

$$\nu \ll \frac{\log s+\log (1+\log h^{1/r})}{\log \log \ s}\ll \frac{\log s+\log (1+\log h^{1/r})}{\log \log \ s}.$$

If $r \gg s^3,$ then

$$\nu \ll 1+ \frac{\log (1+\log h^{1/r})}{\log r}\ll \frac{\log s+\log (1+\log h^{1/r})}{\log s}.$$

Thus,

$$P_{med}(Y_W,Y_S) \ll s \left( \frac{\log s+\log (1+\log h^{1/r})}{\log \log \ s}\right) .$$

We combine the above inequality with (2.1) and Lemma 2.2 to get

$$P(h)\ll s(\log s)h^{2/r} .$$

Using a partial summation argument, it was shown in [8, p. 212] that

$$\begin{aligned} \nonumber N_F(h)\ll P(h)+h^{1/r}r^{-1}\sum _{n=1}^{h-1} P(n)n^{-1-(1/r)}. \end{aligned}$$

Substituting our estimate for P(h), we obtain the result of the theorem.