1 Introduction

Positivity concepts for divisors play a crucial role in algebraic geometry. Among these concepts is ampleness, which can also be interpreted intersection theoretically via the Nakai–Moishezon–Kleiman criterion. A weaker form of positivity is bigness: a divisor D is big iff the growth of the dimension of global sections of its multiples is maximal. The rate of this growth is then measured by the volume of the divisor [35, Sect. 2.1] and for ample divisors this is simply the top self-intersection by the asymptotic Riemann–Roch theorem [35, Theorem 1.1.24]. In [20] Demailly introduces a measure of local positivity of a divisor at a point, the Seshadri constant, in order to study the Fujita conjecture.

The connection between Diophantine approximation and positivity concepts is central to many results on Diophantine geometry. It is a key element in Vojta’s proof of Mordell’s conjecture [51] and in Faltings’s proof of the Mordell–Lang conjecture [25]. In [29] it has been shown that the constants showing up in Diophantine approximations can be obtained as the expectation of certain random variables coming from filtrations on the graded ring of sections of a divisor. Later [18, 22, 23, 27] showed that these constants can be obtained via different geometric invariants. In [39] Diophantine approximation constants are shown to be related to volumes of divisors. This is shown to be true also in the function field case in [30] using an effective Schmidt subspace theorem over function fields [52]. Finally [31, 44, 45] treat the more general case where not only points but closed subschemes are approximated.

Most results on Diophantine approximation rely on the construction of an auxiliary polynomial having a certain order of vanishing at given points. In this paper we present a new approach that follows Faltings’s proof of the Mordell–Lang conjecture [25] using information on local positivity at these points to study the vector spaces of suitable auxiliary polynomials.

One of the most important results in Diophantine approximation is Roth’s theorem on the approximation of algebraic numbers by rationals [43]. It states that for a given algebraic number \(\alpha\) and a given \({\varepsilon }> 0\) there are only finitely many rational numbers \(p/q \in {\mathbb {Q}}\) such that

$$\begin{aligned} \left| \alpha - \frac{p}{q} \right| \le q^{-(2+{\varepsilon })} \,. \end{aligned}$$
(1)

The proof of this theorem consists of two steps:

  1. 1.

    First an auxiliary polynomial \(P \in {\mathbb {Z}}[X_1,\dots ,X_n]\) having a certain order of vanishing at \((\alpha ,\dots ,\alpha )\) is constructed, which is then shown to vanish to a suitable order at \((p_1/q_1,\dots ,p_n/q_n)\) where \(p_i/q_i\) are solutions to (1). Here one usually uses a version of Siegel’s lemma [48].

  2. 2.

    Next, one shows that there exists an upper bound for the order of P at the point \((p_1/q_1,\dots ,p_n/q_n)\) obtaining a contradiction. This upper bound may be either of geometric (Dyson’s lemma [21] or rather its generalization by Esnault and Viehweg [24]) or of arithmetic nature (Roth’s lemma [43] and Faltings’s product theorem [25]).

Note that there are closely related methods in transcendence theory employing a different strategy that does not require Siegel’s lemma, in particular Laurent’s interpolating determinants [34] and Bost’s slope method [11], see also [16].

There are also many results on the simultaneous approximation of algebraic numbers by rationals. The generalization of Roth’s theorem in this context is due to Schmidt [46, Corollary to Theorem 1]. Suppose that \(\alpha _1,\dots ,\alpha _r\) are algebraic numbers such that \(1,\alpha _1,\dots ,\alpha _r\) are linearly independent over \({\mathbb {Q}}\). Then for every \(\varepsilon > 0\) there exist only finitely many r-tuples of rational numbers \((p_1/q,\dots ,p_r/q)\) such that

$$\begin{aligned} \left| \alpha _i - \frac{p_i}{q} \right| \le q^{-(1+1/r+\varepsilon )} \end{aligned}$$
(2)

holds for all \(1\le i\le r\).

The theorems of Roth and Schmidt are not effective in that there is no bound for q for the rational numbers p/q and \(p_i/q\) satisfying (1) and (2) respectively. The earliest effective result in the approximation of a single algebraic number is the theorem of Liouville [36], which is similar to Roth’s theorem with exponent the degree d of the algebraic number in question instead of 2. Fel’dman [26] obtained an improvement of Liouville’s theorem, in which the exponent is strictly smaller than d, however, the difference is extremely small. A different approach to this problem is Bombieri’s Thue–Siegel principle [2, 6,7,8,9,10, 15]. For improvements see for example [5, 13]. In the case of simultaneous approximation there are effective results where the tuple of algebraic numbers is given by rational powers of rational numbers [1, 4, 41, 42].

Here we discuss a different strategy linking methods from positivity and Diophantine approximation that follows Faltings’s proof of the Mordell–Lang conjecture [25]. For a detailed discussion of the strategy of Faltings’s proof see [40]. We consider homogeneous polynomials in two variables having large index at the point \((\alpha _1,\alpha _2)\), see Definition 8, and a priori small index at \((p_1/q,p_2/q)\) where \(p_i/q\) is a suitably good rational approximation of \(\alpha _i\) for \(i=1,2\).

Using Faltings’s Siegel lemma we can then ensure that we can find such a polynomial with suitably bounded coefficients in \({\mathbb {Z}}\). Finally we give a bound for q involving the index of P at \((\alpha _1,\alpha _2)\) and \((p_1/q,p_2/q)\).

The novelty of this approach is that it avoids providing a zero estimate: we only need to suitably bound the dimension of the space of polynomials with given degree and given index at \((\alpha _1,\alpha _2)\), all of its conjugates and \((p_1/q,p_2/q)\). Therefore we only need a partial understanding of the volume function on blowups of \({\mathbb {P}}^2\). The fact that we only consider one solution \((p_1/q,p_2/q)\) will finally make our theorem effective.

We obtain the following theorem.

Theorem 1

Let \(\alpha _1, \alpha _2\) be algebraic numbers and let \(d:=[{\mathbb {Q}}(\alpha _1,\alpha _2):{\mathbb {Q}}]\). Suppose that \((\alpha _1,\alpha _2)\) and all of its conjugates are nonsingular points of an irreducible curve of degree m defined over \({\mathbb {C}}\). Then there exists for all \(\delta \in {\mathbb {Q}}\) with \(\delta > \max \{m,d/m\}\) an effectively computable constant \(C_0(\alpha _1,\alpha _2,\delta ,m)\) depending only on \((\alpha _1,\alpha _2)\), m and \(\delta\) such that for all pairs of rational numbers \((p_1/q,p_2/q)\) satisfying

$$\begin{aligned} \left| \alpha _i - \frac{p_i}{q} \right| \le q^{-\delta } \text { for i=1,2} \end{aligned}$$
(3)

we have \(q \le C_0(\alpha _1,\alpha _2,\delta ,m)\).

The proof of Theorem 1 yields the following corollary on a possible choice for \(C_0(\alpha _1,\alpha _2,\delta ,m)\).

Corollary 2

Using the notation of the previous theorem let \(\alpha _0\) be defined as \(\alpha _1 + M_0 \alpha _2\) where \(M_0\) is the smallest natural number that \(\alpha _1 + M_0 \alpha _2\) is a primitive element of \({\mathbb {Q}}(\alpha _1,\alpha _2)\) (such a \(M_0\) always exists by the proof of the primitive element theorem [33, Theorem V.4.6]). Let \(\alpha\) be defined as \(M_1 \alpha _0\) where \(M_1\) is the smallest natural number such that \(M_1 \alpha _0\) is an algebraic integer. Now let N be the smallest natural number such that \(N \alpha _1\) and \(N \alpha _2\) can be expressed as

$$\begin{aligned} N \alpha _i = c_1^i \alpha ^{d-1} + \dots + c_{d-1}^i \alpha + c_d^i \text { for }i=1,2 \end{aligned}$$

where \(c_h^i \in {\mathbb {Z}}\) and let M be defined as \(\max \{|c_h^i| \,\mid h = 1,\dots ,d \text { and } i=1,2\}\). Let Q be defined as the denominator of

$$\begin{aligned}\theta :=\frac{1/\delta +\min \{1/m,m/d\}}{2}\end{aligned}$$

and let

$$\begin{aligned} \theta _0:= \min \left\{ \frac{\min \{1/m,m/d\}-1/\delta }{4},\frac{1}{Q l(\theta )}\right\} \end{aligned}$$

where

$$\begin{aligned} l(\theta ):= \left\lceil \frac{1}{2} \left\lfloor \frac{((-3+d \theta )Q+4 Q^2 (1-d \theta ^2)+1)^2}{Q^2(1-d \theta ^2)} + 3\right\rfloor \right\rceil . \end{aligned}$$

Then Theorem 1 holds with

$$\begin{aligned} C_0(\alpha _1,\alpha _2,\delta ,m):=\left( 64 (2^{10} (d M (|m_{\alpha }|+1)^d)^{3} )^{1/\theta _0^2} \max \{1,\left| \alpha _1\right| ,\left| \alpha _2\right| \} N^{3} \right) ^{\frac{1}{\delta (\theta - \theta _0)-1}}. \end{aligned}$$

1.1 Notation

In the remainder of this article we will denote by \(\alpha _1\) and \(\alpha _2\) algebraic numbers and let \(d:=[{\mathbb {Q}}(\alpha _1,\alpha _2):{\mathbb {Q}}]\).

2 Seshadri constants on blow-ups of \({\mathbb {P}}^2\)

In this section we will be only concerned with varieties over \({\mathbb {C}}\).

We begin by discussing Seshadri constants. These constants measuring local positivity of divisors were first defined by Demailly in [20] and their name is due to the Seshadri criterion for ampleness [47, Remark 7.1].

Definition 3

Let X be a smooth projective surface, let M be a nef \({\mathbb {R}}\)-divisor on X, let x be a point in X and let \(\pi _x : X' \rightarrow X\) be the blowup of X at x and E its exceptional divisor. Then the Seshadri constant of M at x is defined as

$$\begin{aligned} {\varepsilon }(X,M;x):= \sup \{ t \ge 0 \mid \pi _x^* M - t E \text { is a nef } {\mathbb {R}}- \hbox {divisor on } X'\}. \end{aligned}$$

Let us recall some properties of Seshadri constants.

Lemma 4

([35, Example 5.1.4, Example 5.1.6]) Let \(X,X'\) and x be as above and let M be nef and integral. Then:

  1. 1.

    The Seshadri constant is homogenous:

    $$\begin{aligned} {\varepsilon }(X,l M;x) = l \, {\varepsilon }(X,M;x) \end{aligned}$$

    for all \(l \in {\mathbb {N}}\).

  2. 2.

    If M is very ample then

    $$\begin{aligned} {\varepsilon }(X,M;x) \ge 1 \,. \end{aligned}$$

For more about Seshadri constants the reader may consult [3] and [35, Chapter 5].

We will need the following statement about ample divisors on the blowup of \({\mathbb {P}}^2\) at points that lie on an irreducible curve of degree m.

Proposition 5

Let \(x_1,\dots ,x_{d}\) be distinct points lying on an irreducible curve D of degree m in \({\mathbb {P}}^2\) such that \({{\,\mathrm{mult}\,}}_{x_i}D=1\) for all i, let L be a line in \({\mathbb {P}}^2\) and consider the blow-up \(\pi : S \rightarrow {\mathbb {P}}^2\) of \({\mathbb {P}}^2\) at \(x_1,\dots ,x_{d}\) with exceptional divisors \(E_1,\dots ,E_{d}\). Then for every \(0< t < \min \{1/m,m/d\}\) the \({\mathbb {R}}\)-divisor \(\pi ^* L - t (E_1 + \dots + E_{d})\) is ample.

Proof

The strict transform \(D'\) of D is linearly equivalent to the divisor

$$\begin{aligned} C:=m \, \pi ^* L-(E_1+\dots +E_{d}) \end{aligned}$$

on S. This implies that for \(0 \le t \le 1/m\) we have

$$\begin{aligned} L_t := \pi ^*L - t (E_1 + \dots + E_{d}) = (1-m t) \pi ^* L + t C = (1-m t) \pi ^* L + t (C-D') + t D'. \end{aligned}$$

The intersection of \(\pi ^* L\) and the strict transform of any irreducible curve on \({\mathbb {P}}^2\) is positive and \(C-D'\) is numerically trivial. Further, \(D'\) intersects all irreducible curves on S except possibly itself nonnegatively. Using

$$\begin{aligned} L_t E_i&= t\\ L_{t} D'&= m - d t\\ L_{t}^2&= 1 - d t^2 \end{aligned}$$

we conclude that for \(0< t < \min \{1/m,m/d,1/\sqrt{d}\}\) it holds that \(L_{t}^2>0\) and that \(L_t\) intersects all irreducible curves on S positively. Furthermore, as \(1/m \le m/d\) is equivalent to \(1/m \le 1/\sqrt{d}\), we conclude that the equality \(\min \{1/m,m/d,1/\sqrt{d}\} = \min \{1/m,m/d\}\) holds. By the real version of the Nakai–Moishezon criterion [17] the statement of the proposition holds. \(\square\)

In what follows we will need to have a lower bound for the Seshadri constant of \(\pi ^*L - t (E_1 + \dots + E_{d})\) for \(0 \le t < \min \{1/m,m/d\}\) at another point. In order to do this we will employ an effective version of Matsusaka’s big theorem [37, 38] for surfaces by Fernández del Busto [19]. Note that Siu has given an effective version of Matsusaka’s big theorem valid in higher dimensions [49, 50].

Theorem 6

[19] Let A be an ample divisor on a smooth projective algebraic surface X. Then lA is very ample for every

$$\begin{aligned} l > \frac{1}{2} \left\lfloor \frac{(A(K_X + 4 A)+1)^2}{A^2} + 3 \right\rfloor \,. \end{aligned}$$

Using this we are now ready to prove the following geometric theorem, which will be essential in the proof of the main theorem.

Theorem 7

Let \(x_1,\dots ,x_{d}\) be distinct points lying on an irreducible curve D of degree m in \({\mathbb {P}}^2\) such that \({{\,\mathrm{mult}\,}}_{x_i}D=1\) for all i, let \(x_{d+1} \in {\mathbb {P}}^2\), let L be a line in \({\mathbb {P}}^2\) and consider the blow-up \(\pi : X \rightarrow {\mathbb {P}}^2\) of \({\mathbb {P}}^2\) at \(x_1,\dots ,x_{d+1}\) with exceptional divisors \(E_1,\dots ,E_{d+1}\). Let us define for \(Q>0\)

$$\begin{aligned} l(\theta ):= \left\lceil \frac{1}{2} \left\lfloor \frac{((-3+d \theta )Q+4 Q^2 (1-d \theta ^2)+1)^2}{Q^2(1-d \theta ^2)} + 3\right\rfloor \right\rceil . \end{aligned}$$

Then for all \(\theta \in {\mathbb {Q}}\) with denominator \(Q \in {\mathbb {N}}\) satisfying \(\theta <\min \{1/m,m/d\}\) and for every \(0 \le \mu \le \frac{1}{Q l(\theta )}\) we have that

$$\begin{aligned} {{\,\mathrm{vol}\,}}_{X} (\pi ^* L - \theta (E_1 + \dots + E_{d})) - {{\,\mathrm{vol}\,}}_{X} (\pi ^*L - \theta (E_1 + \dots + E_{d}) - \mu E_{d+1}) = \mu ^2\,. \end{aligned}$$

Proof

By Proposition 5 above we know that \(M:=(\pi ^*L - \theta (E_1 + \dots + E_{d}))\) is ample. Note that \(l(\theta ) = \left\lceil \frac{1}{2} \left\lfloor \frac{(Q M(K_X + 4 Q M)+1)^2}{(Q M)^2} + 3 \right\rfloor \right\rceil\). By Theorem 6 and because QM is an integral ample divisor, we now know that the divisor \(l(\theta ) Q M\) is very ample and therefore \({\varepsilon }(\text {Bl}_{x_1,\dots ,x_{d}}({\mathbb {P}}^2),M; x_{d+1}) \ge \frac{1}{Q l(\theta )}\) by the properties of Seshadri constants in Lemma 4 (note that the homogenity of Seshadri constants immediately extends to \({\mathbb {Q}}\)-divisors). Therefore \(\pi ^* L - \theta (E_1 + \dots + E_{d})-\mu E_{d+1}\) is nef for \(0 \le \mu \le \frac{1}{Q l(\theta )}\) and the statement of the theorem follows by the asymptotic Riemann–Roch theorem [35, Corollary 1.4.41]. \(\square\)

3 Bounding the denominator of a good approximation

In this section we will bound the denominator q of a good approximation \((p_1/q,p_2/q)\) of \((\alpha _1,\alpha _2)\) using a polynomial \(P \in {\mathbb {Z}}[X_1,X_2]\) with suitably bounded coefficients and suitable index at \((\alpha _1,\alpha _2)\) and \((p_1/q,p_2/q)\). This chapter closely follows [32, §D.5]. For the convenience of the reader we give proofs as we need slightly different statements than those in [32].

Definition 8

Let

$$\begin{aligned} P = \sum _{j \in {\mathbb {N}}_0^2} \, a_{j_1,j_2} {X_1}^{j_1} {X_2}^{j_2} \end{aligned}$$

be a polynomial with coefficients in \({\mathbb {C}}\).

  1. 1.

    For a multi-index \(j \in {\mathbb {N}}_0^2\) we define a differential operator \(\partial _j\) via

    $$\begin{aligned}\partial _j P := \frac{1}{j_1! j_2!} \frac{\partial ^{j_1+j_2}}{\partial X_1^{j_1} \partial X_2^{j_2}} P .\end{aligned}$$
  2. 2.

    If \(P \ne 0\), we define the index of P at \(x=(x_1,x_2)\) with respect to the weights \((r_1,r_2) \in {\mathbb {N}}^2\) to be the nonnegative real number

    $$\begin{aligned}{{\,\mathrm{ind}\,}}_{(x_1,x_2;r_1,r_2)}(P) := \min \{ j_1/r_1+j_2/r_2 \mid j \in {\mathbb {N}}_0^2, \partial _j P (x) \ne 0\}.\end{aligned}$$
  3. 3.

    If \(P \in {\mathbb {Z}}[X_1,X_2]\), the naive height of P is defined as

    $$\begin{aligned}|P| := max \{ |a_j| \,\mid j \in {\mathbb {N}}_0^2\} \,.\end{aligned}$$

Let us summarize some properties of the index and the differential operators \(\partial _j\) for later use.

Lemma 9

([32, Lemmas D.3.1 and D.3.2]) Let \(P \in {\mathbb {C}}[X_1,X_2]\) and let \(j \in {\mathbb {N}}_0^2\).

  1. 1.

    If \(\partial _j P \ne 0\), then \({{\,\mathrm{ind}\,}}_{(x_1,x_2;r_1,r_2)}(\partial _{j} P) \ge {{\,\mathrm{ind}\,}}_{(x_1,x_2;r_1,r_2)}(P) - j_1/r_1-j_2/r_2\),

  2. 2.

    if \(P \in {\mathbb {Z}}[X_1,X_2]\), we have \(\partial _{j} P \in {\mathbb {Z}}[X_1,X_2]\),

  3. 3.

    if \(\deg P \le k\), then \(|\partial _{j} P| \le 4^{k} |P|\).

From now on we will make the assumption \(r_1 = r_2 = k\). In particular we have that \({{\,\mathrm{ind}\,}}_{(\alpha _1, \alpha _2; k, k)}(P) = {{\,\mathrm{ord}\,}}_{(\alpha _1, \alpha _2)}(P)/k\).

In the following two lemmas we will provide a bound on the denominator and absolute value of derivatives of a polynomial \(P \in {\mathbb {Z}}[X_1,X_2]\) at \((p_1/q,p_2/q)\).

Lemma 10

Let \(P \in {\mathbb {Z}}[X_1,X_2]\) be a polynomial of degree less or equal k and let \(j \in {\mathbb {N}}_0^2\) be a multi-index. Then

$$\begin{aligned} q^k \partial _j P(p_1/q,p_2/q) \in {\mathbb {Z}}\,.\end{aligned}$$

Proof

Using Lemma 9 we obtain that the the coefficients of \(\partial _j P\) are in \({\mathbb {Z}}\). Therefore \(\partial _j P(p_1/q,p_2/q)\) is a sum of terms whose denominators are divisors of \(q^{k}\) giving us the desired bound. \(\square\)

Lemma 11

Let \(P \in {\mathbb {Z}}[X_1,X_2]\) of degree less or equal k with \(k \ge 4\) and let \(j \in {\mathbb {N}}_0^2\) be a multi-index. Let \(\theta\) be the index of P at \((\alpha _1,\alpha _2)\) with respect to (k, k), let \(0<\theta _0<\theta\), let \(\delta >0\) and let \(N \in {\mathbb {N}}\).

Then it holds that for \((p_1/q,p_2/q) \in {\mathbb {Q}}\) satisfying

$$\begin{aligned} \left| \alpha _i - \frac{p_i}{q} \right| \le N q^{-\delta } \hbox { for}\ i=1,2 \end{aligned}$$
(4)

and for every \(j=(j_1,j_2) \in {\mathbb {N}}_0^2\) such that \(\frac{j_1+j_2}{k} \le \theta _0\) we have

$$\begin{aligned} \left| \partial _j P(p_1/q,p_2/q) \right| \le 64^k \left| P \right| (\max \{1,\left| \alpha _1\right| ,\left| \alpha _2\right| \})^{k} N^{2 k} q^{-k \delta (\theta - \theta _0)}.\end{aligned}$$

Proof

The claim of the lemma is evident for \(\partial _j P = 0\). We may therefore assume that \(\partial _j P \ne 0\). First note that for all \(i \in {\mathbb {N}}_0^2\) we have that \(\partial _i\partial _j P(\alpha _1,\alpha _2)\) is a sum of at most \(1/2 \, (k+1) (k+2) \le 2^k\) terms because \(k \ge 4\). These terms are of the form \(c_{i_1,i_2} \alpha _1^{i_1}\alpha _2^{i_2}\) with \(c_{i_1,i_2} \in {\mathbb {Z}}\) by Lemma 9 and \(i_1+i_2 \le k\) and are themselves bounded by

$$\begin{aligned}\left| c_{i_1,i_2} \alpha _1^{i_1}\alpha _2^{i_2} \right| \le \left| \partial _i \partial _j P \right| (\max \{1,\left| \alpha _1\right| ,\left| \alpha _2\right| \})^{k} \le 16^k |P| (\max \{1,\left| \alpha _1\right| ,\left| \alpha _2\right| \})^{k}\end{aligned}$$

where we have used Lemma 9 two times. We may now expand \(\partial _j P\) around \((\alpha _1,\alpha _2)\) and use that Lemma 9 implies \({{\,\mathrm{ind}\,}}_{(\alpha _1,\alpha _2;k,k)}(\partial _j P) \ge \theta - \theta _0\) to obtain

$$\begin{aligned} \partial _j P (p_1/q,p_2/q) = \sum _{\begin{array}{c} 0 \le i_1,i_2 \le k \\ \theta - \theta _0 \le (i_1+i_2)/k \le 1 \end{array}} (\partial _i \partial _j P)(\alpha _1,\alpha _2) (p_1/q - \alpha _1)^{i_1} (p_2/q - \alpha _2)^{i_2} \end{aligned}$$

and by assumption (4), the fact that the number of terms above is bounded by \(2^k\) and the bounds above we have

$$\begin{aligned} \left| \partial _j P (p_1/q,p_2/q) \right| \le 64^k |P| (\max \{1,\left| \alpha _1\right| ,\left| \alpha _2\right| \})^{k} N^{2 k} q^{-k \delta (\theta - \theta _0)} \,. \end{aligned}$$

\(\square\)

Using the results above we obtain a bound for the denominator of a good approximation as follows.

Lemma 12

Let \(k \ge 4\) be a positive integer. Let \(0< \theta _0 < \theta\) be given and suppose that \((p_1/q,p_2/q) \in {\mathbb {Q}}^2\) is a solution of inequality (4) for given \(\delta > 1/(\theta - \theta _0)\), \(N \in {\mathbb {N}}\). Now assume that \(P \in {\mathbb {Z}}[X_1,X_2]\) satisfies the following properties:

  1. 1.

    the degree of P is at most k,

  2. 2.

    the index of P at \((\alpha _1,\alpha _2)\) with respect to the weights (k, k) satisfies

    $$\begin{aligned}{{\,\mathrm{ind}\,}}_{(\alpha _1,\alpha _2;k,k)}(P) \ge \theta ,\end{aligned}$$
  3. 3.

    \(|P| \le B^{k}\), where B depends only on \((\alpha _1,\alpha _2)\), k and \(\delta\).

Let

$$\begin{aligned}C(\alpha _1,\alpha _2,\delta ,N) :=\left( 64 B \max \{1,\left| \alpha _1\right| ,\left| \alpha _2\right| \} N^{2} \right) ^{\frac{1}{\delta (\theta - \theta _0)-1}} \,.\end{aligned}$$

Then it holds that if

$$\begin{aligned} {{\,\mathrm{ind}\,}}_{(p_1/q, p_2/q; k, k)}(P) < \theta _0\,, \end{aligned}$$

we have \(q \le C(\alpha _1,\alpha _2,\delta ,N)\).

Proof

Assume \({{\,\mathrm{ind}\,}}_{(p_1/q, p_2/q; k, k)}(P) < \theta _0\) and let \(j \in {\mathbb {N}}_0^2\) with \(\frac{j_1+j_2}{k} < \theta _0\) be such that \(\partial _j P(p_1/q, p_2/q) \ne 0\), say \(\partial _j P(p_1/q, p_2/q) = s/m\) with \(s \in {\mathbb {Z}}\setminus \{0\}, m \in {\mathbb {N}}\) and s and m coprime. Now Lemma 11 and the bound on |P| give us that

$$\begin{aligned} |\partial _j P(p_1/q,p_2/q)| \le \left( 64 B \max \{1,\left| \alpha _1\right| ,\left| \alpha _2\right| \} N^{2} q^{-\delta (\theta - \theta _0)}\right) ^k \,. \end{aligned}$$

We use the principle that there is no integer strictly between 0 and 1 to obtain

$$\begin{aligned} 1/m \le \left( 64 B \max \{1,\left| \alpha _1\right| ,\left| \alpha _2\right| \} N^{2} q^{-\delta (\theta - \theta _0)}\right) ^k. \end{aligned}$$

Finally Lemma 10 gives

$$\begin{aligned} q^{-k} \le \left( 64 B \max \{1,\left| \alpha _1\right| ,\left| \alpha _2\right| \} N^{2} q^{-\delta (\theta - \theta _0)}\right) ^k \end{aligned}$$

and after taking k-th roots and simplifying we obtain

$$\begin{aligned} q^{\delta (\theta - \theta _0)-1} \le 64 B \max \{1,\left| \alpha _1\right| ,\left| \alpha _2\right| \} N^{2} \end{aligned}$$

and the claimed inequality follows. \(\square\)

4 Finding a suitable global section

For this section we fix an embedding \({\mathbb {A}}^2 \hookrightarrow {\mathbb {P}}^2\) and consider the line \(L = {\mathbb {P}}^2 \setminus {\mathbb {A}}^2\) such that the global sections of \({\mathcal {O}}_{{\mathbb {P}}^2}(k L)\) restricted to \({\mathbb {A}}^2\) are the polynomials of degree less or equal k, and view \((\alpha _1,\alpha _2)\), all of its conjugates and \((p_1/q,p_2/q)\) as elements of \({\mathbb {P}}^2\) via this embedding. In this chapter we will always indicate which base field we are working over.

We now state Faltings’s version of Siegel’s lemma.

Lemma 13

([25, Proposition 2.18]) Let V, W be two finite dimensional normed \({\mathbb {R}}\)-vector spaces and let \(M \subset V\) and \(N \subset W\) be \({\mathbb {Z}}\)-lattices of maximal rank. Let further \(\phi : V \rightarrow W\) be a linear map such that \(\phi (M) \subset N\). Let \(b:=\dim (V)\) and \(a:=\dim ({{\,\mathrm{Ker}\,}}(\phi ))\) and assume that there exists a constant \(C \ge 2\) such that

  1. 1.

    M is generated by elements of norm at most C,

  2. 2.

    the norm of \(\phi\) is bounded by C,

  3. 3.

    all non-trivial elements of M and N have norm at least 1/C.

For \(1 \le i \le b\) set

$$\begin{aligned} \lambda _i := \inf \{\lambda > 0 \mid \exists i \hbox {linearly independent vectors of norm} \le \lambda \hbox { in } {{\,\mathrm{Ker}\,}}(\phi )\cap M \} \,. \end{aligned}$$

Then it holds that

$$\begin{aligned} \lambda _{i+1} \le (C^{3b} b!)^{1/(a-i)}\,. \end{aligned}$$

We will need the following number theoretical lemma.

Lemma 14

([32, Lemma D.3.4]) Let \(\alpha \in {\overline{{\mathbb {Q}}}}\) be an algebraic integer of degree \(d_{\alpha }:=[{\mathbb {Q}}(\alpha ):{\mathbb {Q}}]\) over \({\mathbb {Q}}\) and let \(m_{\alpha } \in {\mathbb {Q}}[X]\) be the minimal polynomial of \(\alpha\) over \({\mathbb {Q}}\). Then we have \(\alpha ^l = a_1^{(l)} \alpha ^{d_{\alpha }-1}+\dots +a_{d_{\alpha }}^{(l)}\) with \(a_i^{(l)} \in {\mathbb {Z}}\) satisfying \(\left| a_i^{(l)} \right| \le (|m_{\alpha }|+1)^l\).

The following Lemma now clarifies how we intend to use Faltings’s version of Siegel’s lemma. In it we will make an assumption that implies that \(\alpha _1,\alpha _2\) are algebraic integers. Note that we can always satisfy this assumption by considering \(N \alpha _i\) instead of \(\alpha _i\) for a suitable \(N \in {\mathbb {N}}\).

Lemma 15

Let \(k \ge 4\) be a positive integer, let \(B_k:=H^0({\mathcal {O}}_{{\mathbb {P}}^2_{\mathbb {Q}}}(k L))\), which we will identify with the polynomials of degree less or equal k in \({\mathbb {Q}}[X_1,X_2]\), and let \(A_k\) be the subspace of sections whose index at \((\alpha _1,\alpha _2)\) with respect to the weights (k, k) is at least \(\theta\). Choose an algebraic integer \(\alpha\) which is a primitive element for \({\mathbb {Q}}(\alpha _1,\alpha _2)\) and assume that \(\alpha _1\) and \(\alpha _2\) can be expressed as

$$\begin{aligned} \alpha _i = c_1^i \alpha ^{d-1} + \dots + c_{d-1}^i \alpha + c_d^i \hbox { for}\ i=1,2 \end{aligned}$$

where \(c_h^i \in {\mathbb {Z}}\) and let M be defined as \(\max \{|c_h^i| \, \mid h = 1,\dots ,d \text { and } i=1,2\}\). Then there exists a linear map \(\phi _k : B_k \otimes {\mathbb {R}}\rightarrow {\mathbb {Q}}(\alpha _1,\alpha _2)^{l_k} \otimes {\mathbb {R}}\) where

$$\begin{aligned} l_k= \#\{j \in {\mathbb {N}}_0^2 \mid \frac{j_1+j_2}{k} < \theta \} \end{aligned}$$

such that

  1. 1.

    \({{\,\mathrm{Ker}\,}}(\phi _k)=A_k \otimes {\mathbb {R}}\),

  2. 2.

    \(\phi _k\), the lattice inside \(B_k \otimes {\mathbb {R}}\) generated by monomials and the lattice in \({\mathbb {Q}}(\alpha _1,\alpha _2)^{l_k} \otimes {\mathbb {R}}\) generated by \(\alpha ^i\) for \(i=0,\dots ,d-1\) in every component satisfy the conditions in Lemma 13 with \(C=B^{k}\) where \(B>0\) is the following constant

    $$\begin{aligned} B:= 8 d M (|m_{\alpha }|+1)^d \,. \end{aligned}$$

Proof

Define the linear map

$$\begin{aligned} \phi _k : B_k \otimes {\mathbb {R}}&\rightarrow {\mathbb {Q}}(\alpha _1,\alpha _2)^{l_k} \otimes {\mathbb {R}}\\ P \otimes 1&\mapsto (\partial _j P) (\alpha _1,\alpha _2) \otimes 1 \end{aligned}$$

where j ranges over all pairs of non-negative integers satisfying \((j_1+j_2)/k < \theta\). Consider the basis of \(V:=B_k \otimes {\mathbb {R}}\) which consists of monomials, the basis of \(W:={\mathbb {Q}}(\alpha _1,\alpha _2)^{l_k} \otimes {\mathbb {R}}\) consisting of \(\alpha ^i\) for \(i=0,\dots ,d-1\) in every component and the lattices \(L_V\) and \(L_W\) generated by these bases.

By Lemma 9 and the assumptions on \(\alpha\) we have that \(\phi _k(L_V) \subset L_W\). We now identify \(V \cong {\mathbb {R}}^{\dim _{\mathbb {Q}}B_k}\) and \(W \cong {\mathbb {R}}^{d l_k}\) using the above bases and equip these \({\mathbb {R}}\)-vector spaces with the maximum norm \(|\cdot |_{\infty }\). It is then clear that \(L_V\) is generated by elements of norm 1 and all non-trivial elements of \(L_V\) and \(L_W\) have norm greater or equal 1. Therefore we only need to give a bound on the norm of \(\phi _k\).

To achieve this, we consider a polynomial \(P \in V\), note that P is a sum of at most \(1/2 \, (k+1) (k+2) \le 2^k\) terms and use Lemma 9 to obtain that the coefficients of P are bounded by \(|\partial _j P| \le 4^k|P|\). Then by using the assumptions of Lemma 15 to expand \(\alpha _1^u \alpha _2^v\) where \(u+v \le k\) into a \({\mathbb {Z}}\)-linear combination of powers of \(\alpha\) we obtain a sum of \(d^{u+v} \le d^{k}\) terms \(R \alpha ^l\) with \(l\le (u+v)d \le k d\) and \(R \le M^{u+v} \le M^{k}\). By Lemma 14 we have that \(\alpha ^l\) is then a \({\mathbb {Z}}\)-linear combination of \(1,\alpha ,\dots ,\alpha ^{d-1}\) with coefficients bounded by \((|m_{\alpha }|+1)^{k d}\). Therefore it holds that

$$\begin{aligned} |\phi _k(P) |_{\infty } \le |P| (8 d M (|m_{\alpha }|+1)^d)^{k} \end{aligned}$$

and this implies the statement of the lemma. \(\square\)

Lemma 16

Let us keep the notation and assumptions of the previous lemma and let \(\pi : X_{\mathbb {C}}\rightarrow {\mathbb {P}}^2\) be the blowup of \({\mathbb {P}}^2_{\mathbb {C}}\) in \((\alpha _1,\alpha _2)\) and all of its conjugates with corresponding exceptional divisors \(E_1,\dots ,E_d\) and in \((p_1/q,p_2/q)\) with corresponding exceptional divisor \(E_{d+1}\). Letting \(b_k:= \dim _{\mathbb {Q}}B_k\), \(a_k:= \dim _{\mathbb {Q}}A_k\), \(i_k:= \dim _{\mathbb {Q}}U_k\) where \(U_k\) is the linear subspace of \(A_k\) of sections \(s \in A_k\) with \({{\,\mathrm{ind}\,}}_{(p_1/q,p_2/q;k,k)} s \ge \theta _0\) we have that

$$\begin{aligned} \lim _{k \rightarrow \infty } \frac{b_k}{k^2/2}&= {{\,\mathrm{vol}\,}}_{{\mathbb {P}}^2_{\mathbb {C}}}(\pi ^*L) = 1 \, \\ \lim _{k \rightarrow \infty } \frac{a_k}{k^2/2}&= {{\,\mathrm{vol}\,}}_{X_{\mathbb {C}}}(\pi ^*L - \theta \, (E_1 + \dots + E_d))\\ \lim _{k \rightarrow \infty } \frac{i_k}{k^2/2}&= {{\,\mathrm{vol}\,}}_{X_{\mathbb {C}}}(\pi ^*L - \theta \, (E_1 + \dots + E_d) - \theta _0 E_{d+1}). \end{aligned}$$

Proof

For the first statement note that \(b_k = 1/2\, (k+1)(k+2)\).

Regarding the second and third statement note that \(\theta\) and \(\theta _0\) are real numbers and that the volume function for real divisors is defined by extending the volume function on \({\mathbb {Q}}\)-divisors [35, Corollary 2.2.45]. However by [28, Theorem 3.5] it holds that for a \({\mathbb {R}}\)-Cartier \({\mathbb {R}}\)-divisor D on a projective variety V we have

$$\begin{aligned} {{\,\mathrm{vol}\,}}_V(D) = \lim _{k \rightarrow \infty } \frac{h^0(\lfloor k D\rfloor )}{k^{\dim (V)}/\dim (V)!}\,. \end{aligned}$$

The vector space \(H^0({\mathcal {O}}_{X_{\mathbb {C}}}(\lfloor k \pi ^*L - k \theta \, (E_1 + \dots + E_d) \rfloor ))\) is the space of complex polynomials of degree k vanishing at \((\alpha _1,\alpha _2)\) and all of its conjugates with multiplicity at least \(\lfloor k \theta \rfloor\). We may view this space as the linear subspace of \({\mathbb {C}}^{b_k}\cong H^0({\mathcal {O}}_{X_{\mathbb {C}}}(k \pi ^*L))\) given as the solution set of the equations \(\partial _j P (\alpha _1,\alpha _2) = 0\) where \(P \in H^0({\mathcal {O}}_{X_{\mathbb {C}}}(k \pi ^*L))\) and j ranges over all pairs of non-negative integers satisfying \((j_1+j_2)/k < \theta\). The coefficients of these equations are algebraic and therefore there exists a basis consisting of algebraic elements of \({\mathbb {C}}^{b_k}\) and the dimension is equal to the dimension of the solution set of the same equations in \({\overline{\mathbb {Q}}}^{b_k}\). The absolute Galois group \(G_{\mathbb {Q}}\) of \({\mathbb {Q}}\) acts on the coefficient vectors by permutating them and therefore the solution space in \({\overline{\mathbb {Q}}}^{b_k}\) is stable under \(G_{\mathbb {Q}}\). By [12, Corollary on page V.63] the dimension of this space equals the dimension of the solution set intersected with \({\mathbb {Q}}^{b_k}\) and this number is equal to \(a_k\). The same argument yields \(\dim _{\mathbb {C}}H^0({\mathcal {O}}_{X_{\mathbb {C}}}(\lfloor k \pi ^*L - k \theta \, (E_1 + \dots + E_d) -k \theta _0 \, E_{d+1}\rfloor ))=i_k\) and the statement of the lemma follows. \(\square\)

5 Proof of the main theorem

We are now ready to conclude the proof of the main theorem. In this section we will use the notation of Theorem 1 and Corollary 2.

Proof of the main theorem

Let us consider the asymptotics obtained in the Lemma 16 above and use Faltings’s version of Siegel’s lemma. In order to use Lemma 15 we replace \(\alpha _i\) by \(N \alpha _i\). After this replacement we have that

$$\begin{aligned} \left| N \alpha _i - \frac{N p_i}{q} \right| \le N q^{-\delta } \text { for i=1,2}\,. \end{aligned}$$

We have by Lemma 15 that \(B^{k}\) satisfies the assumptions on C in Lemma 13 and therefore

$$\begin{aligned} \lambda _{i_k+1} \le (( 8 d M (|m_{\alpha }|+1)^d)^{3 k b_k} b_k!)^{1/(a_k-i_k)} \le ((8 d M (|m_{\alpha }|+1)^d)^{3 k} b_k)^{b_k/(a_k-i_k)} \,. \end{aligned}$$
(5)

By the choice of \(\theta\) and \(\theta _0\), Lemma 16 and Theorem 7 the exponent on the right hand side of (5) satisfies

$$\begin{aligned}&\lim _{k \rightarrow \infty } \frac{b_k}{(a_k-i_k)}\\&\quad = \frac{{{\,\mathrm{vol}\,}}_{{\mathbb {P}}^2}(L)}{{{\,\mathrm{vol}\,}}_X(L - \theta (E_1 + \dots + E_d)) - {{\,\mathrm{vol}\,}}_X(L - \theta (E_1 + \dots + E_d) - \theta _0 E_{d+1})}\\&\quad = 1/\theta _0^2 \,. \end{aligned}$$

Now Lemma 13 shows the existence of \(i_k+1\) linearly independent elements of \(A_k\) such that their norm is bounded by

$$\begin{aligned}((8 d M (|m_{\alpha }|+1)^d)^{3 k} b_k)^{b_k/(a_k-i_k)} \le (2^{10} (d M (|m_{\alpha }|+1)^d)^{3} )^{k b_k/(a_k-i_k)}\end{aligned}$$

where we have used that \(b_k=1/2 \, (k+1) (k+2) \le 2^k\). In particular, for \(k\gg 0\) at least one of those elements P is not an element of \(U_k\). Noting that

$$\begin{aligned}\delta (\theta - \theta _0) \ge 1/4 \, \delta \min \{1/m,m/d\} +3/4 > 1\,,\end{aligned}$$

we conclude that P satisfies all of the conditions for Lemma 12 and we obtain that

$$\begin{aligned} q \le \left( 64 (2^{10} (d M (|m_{\alpha }|+1)^d)^{3} )^{b_k/(a_k-i_k)} \max \{1,\left| \alpha _1\right| ,\left| \alpha _2\right| \} N^{3} \right) ^{\frac{1}{\delta (\theta - \theta _0)-1}}\,. \end{aligned}$$

Finally we take the limit for \(k \rightarrow \infty\) and obtain

$$\begin{aligned} q \le \left( 64 (2^{10} (d M (|m_{\alpha }|+1)^d)^{3} )^{1/\theta _0^2} \max \{1,\left| \alpha _1\right| ,\left| \alpha _2\right| \} N^{3} \right) ^{\frac{1}{\delta (\theta - \theta _0)-1}}\,, \end{aligned}$$

which finishes the proof. \(\square\)

Remark 17

The effective constant of Corollary 2 is likely not close to being optimal. On the one hand there exist versions of Siegel’s lemma like [14] that are not directly applicable in the situation of Lemma 15, but have a better dependence on the involved quantities. This suggests that an improvement of Lemma 15 might be possible. On the other hand one might hope that there exist better lower bounds on Seshadri constants than the ones used in the proof of Theorem 7. Both improvements would yield a better estimate for the effective constant of Corollary 2.