Diophantine approximation by conjugate algebraic integers

Damien Roy and Michel Waldschmidt

Département de Mathématiques Université d’Ottawa, Canada Université P. et M. Curie (Paris VI) Institut de Mathématiques, Paris, France

1. Introduction. In their 1969 seminal paper [4], Davenport and Schmidt investigated the approximation of an arbitrary fixed real number ξ by algebraic integers of bounded degree. They proved that if n ≥ 3 is an integer and if ξ is not algebraic over Q of degree at most (n − 1)/2 then there are infinitely many algebraic integers α of degree at most n such that

|ξ − α| ≤ cH(α)^{−[(n+1)/2]}

where c is a positive constant depending only on n and ξ and where H(α) denotes the usual height of α, that is the maximum absolute value of the coefficients of its irreducible polynomial over Z. They also provided refinements for n ≤ 4. Recently, Bugeaud and Teulié revisited this result and showed in [2] that we may also impose that all approximations α have degree exactly n over Q.

Here we establish a similar result for simultaneous approximation by several conjugate algebraic integers. In order to cover the case where ξ is a complex or p-adic number, we will assume more generally that ξ belongs to the completion of a number field K at some place w.

Thus, we fix an algebraic extension K of Q of finite degree d. For each place v of K, we denote by K_{v} the completion of K at v and by d_{v} its local degree at v. We also normalize the corresponding absolute value | |_{v} as in [3] by asking that, when v is above a prime number p of Q, we have |p|_{v} = p^{−d_{v}/d} and that, when v is an Archimedean place, we have |x|_{v} = |x|^{d_{v}/d} for any x ∈ Q. Then, our result of approximation reads as follows:

Theorem A. Let n and t be integers with 1 ≤ t ≤ n/4. Let w be a place of K and let ξ be an element of K_{w} which is not algebraic over K of degree ≤ (n + 1)/(2t). Assume further that |ξ|_{w} ≤ 1 if w is ultrametric. Then there exist infinitely many algebraic integers α which have degree n + 1 over K and admit (over K) t distinct conjugates α_{1}, …, α_{t} in

2000 Mathematics classification. Primary 11J13; Secondary 11J61, 11J82 Work of first author partially supported by NSERC and CICMA

Approximation by conjugate integers

K_{w} satisfying

max_{1≤i≤t} |ξ − α_{i}|_{w} ≤ cH(α)^{−(n+1)/(4dt^{2})}  (1.1)

where c is a constant depending only on K, n, w and ξ.

Note that, for t = 1, K = Q and K_{w} = R, this result is comparable to Theorem 2 of [4] mentioned above (with a shift of 1 in the degree of the approximation). Note also that, if w is ultrametric, any algebraic integer α in K_{w} satisfies |α|_{w} ≤ 1 and so the condition |ξ|_{w} ≤ 1 is necessary to approximate ξ by such numbers.

In section §10, we show that the exponent of approximation (n + 1)/(4dt^{2}) in (1.1) is essentially best possible. In answer to a question of K. Tishchenko, we also show that one cannot hope a similar exponent for simultaneous approximation of t numbers. Taking K = Q and K_{w} = R, we prove a result which implies that, if 2 ≤ t ≤ n, then there exist a constant c > 0 and t real numbers ξ_{1}, …, ξ_{t} such that

max_{1≤i≤t} |ξ_{i} − α_{i}| ≥ cH(α)^{−3n^{1/t}}

for any choice of t distinct conjugates α_{1}, …, α_{t} ∈ C of an algebraic number α of degree between t and n.

The proof of Theorem A uses the same general strategy as Davenport and Schmidt in [4]. It relies on a duality argument combined with the following version of Gel’fond’s criterion of algebraic independence where, for a polynomial Q ∈ K[T], an integer j ≥ 0 and a place v of K, the notation ∥Q∥_{v} stands for the maximal v-adic absolute value of the coefficients of Q, while Q^{(j)} denotes the j-th derivative of Q: Theorem B. Let k, n and t be integers with 1 ≤ t ≤ k ≤ n/4. Let w be a place of K and let ξ be an element of K_{w}. There exists a constant c > 0 which depends only on K, n, w and ξ and has the following property. Assume that, for each sufficiently large real number X, there exists a nonzero polynomial Q ∈ K[T] of degree at most n which satisfies ∥Q∥_{v} ≤ 1 for each place v of K distinct from w and also

max_{0≤j≤n−t} |Q^{(j)}(ξ)|_{w} ≤ cX^{−t/(k+1−t)} and max_{n−t<j≤n} |Q^{(j)}(ξ)|_{w} ≤ X.  (1.2)

Then, ξ is algebraic over K of degree ≤ (n − k + 1)/(2t).

Note that Theorem B may still hold with an exponent smaller than t/(k +1−t) in the above conditions (1.2). However, we will see in §3 that it would not hold with an exponent smaller than t/(n + 1 − t).

Approximation by conjugate integers

It is also interesting to compare this result with the criterion of algebraic independence with multiplicities of [10]. A main difference is that the above theorem requires that a large proportion of the derivatives of the polynomials Q at ξ are small (at least three quarters of them), while the conditions in Proposition 1 of [10] are sharp only when a small proportion of these derivatives are taken into account (say, at most the first half of them).

This paper is organized as follows. The next section §2 sets the various notions of heights that we use throughout the paper. The results of duality which are needed to deduce Theorem A from Theorem B are given in §3, but the proof of this implication is postponed to §9. Sections §§4–7 are devoted to preliminary results towards the proof of Theorem B which is completed in §8. In particular, we establish in §4 a version of Gel’fond’s criterion (without multiplicities) which includes Theorem 2b of [4] and §5 presents a height estimates which generalizes Theorem 3 of [4]. We conclude in §10 with two remarks on the exponent of approximation in Theorem A.

Notation. Throughout this paper, n denotes a positive integer, w denotes a place of K and ξ an element of K_{w}. For shortness, we will sometimes use the expressions a ≪ b or b ≫ a to mean that given non-negative real numbers a and b satisfy a ≤ cb for some positive constant c which depends only on K, n, w and ξ. Overall, we tried to be coherent with the notation of [4].

2. Heights.

Recall that K is a fixed algebraic extension of Q of degree d. With the normalization of its absolute values given in the introduction, the product formula reads

∏_{v} |a|_{v} = 1

for any non-zero element a of K.

Let n be a positive integer. For any place v of K and any n-tuple \underline{a} = (a_{1}, …, a_{n}) ∈ K_{v}^{n}, we define the norm of \underline{a} as its maximum norm ∥\underline{a}∥_{v} = max\{|a_{1}|_{v}, …, |a_{n}|_{v}\}. Accordingly, the (absolute) height of a point \underline{a} of K^{n} is defined by

H(\underline{a}) = ∏_{v} ∥\underline{a}∥_{v}.

If m is a positive integer with 1 ≤ m ≤ n and if M is an m×n matrix with coefficients in K_{v} for some place v of K, we define ∥M∥_{v} as the norm of the \binom{n}{m}-tuple formed by the

Approximation by conjugate integers

minors of order m of M in some order. When M has coefficients in K, we define H(M) as the height of the same point. In particular, for an m × n matrix M of rank m with coefficients in K we have H(M) ≥ 1.

If V is a subspace of K^{n} of dimension m ≥ 1, we define the height H(V) of V to be the height of a set of Plücker coordinates of V. In other words, we define H(V) = H(M) where M is an m × n matrix whose rows form a basis of V. This is independent of the choice of M. According to a well-known duality principle, we also have H(V) = H(N) where N is any (n − m) × n matrix such that V is the solution-set \{\underline{a} ∈ K^{n} ; N\underline{a} = 0\} of the homogeneous system attached to N (see the formula (4), page 433 of [14]). When V = 0, we set H(V) = 1.

We denote by E_{n} the subspace of K[T] consisting of all polynomials of degree ≤ n, and for each place v of K, we denote by E_{n,v} the closure of E_{n} in K_{v}[T]. We also identify E_{n} with K^{n+1} and E_{n,v} to K_{v}^{n+1} by mapping a polynomial a_{0} + a_{1}T + ⋯ + a_{n}T^{n} to the (n + 1)-tuple (a_{0}, …, a_{n}) of its coefficients. Accordingly, we define the norm ∥P ∥_{v} of a polynomial P ∈ E_{n,v} as the maximum of the absolute values of its coefficients and the height H(P) of a polynomial P ∈ E_{n} as the height of the (n + 1)-tuple of its coefficients. In the sequel, we will repeatedly use the fact that, if P_{1}, …, P_{s} ∈ K[T] have product P = P_{1} ⋯ P_{s} of degree ≤ n, then we have

e^{−n}H(P_{1}) ⋯ H(P_{s}) < H(P) < e^{n}H(P_{1}) ⋯ H(P_{s})

as one gets for instance by comparing ∥P_{1}∥_{v} ⋯ ∥P_{s}∥_{v} and ∥P ∥_{v} at all places v of K using the various estimates of §2 in Chapter 3 of [9].

3. Duality.

In this section, we fix a positive integer n, a place w of K and an element ξ of K_{w}. We define below a family of adelic convex bodies and establish about them a result of duality that we will need to deduce Theorem A from Theorem B. We also describe consequences of the adelic Minkowski’s theorem of Macfeat [13] and Bombieri and Vaaler [3] for this type of convex.

For any (n + 1)-tuple of positive real numbers \underline{X} = (X_{0}, X_{1}, …, X_{n}), we define an adelic convex body

C(\underline{X}) = ∏_{v} C_{v}(\underline{X}) ⊂ ∏_{v} E_{n,v}

Approximation by conjugate integers

by putting

C_{w}(\underline{X}) = \{P ∈ E_{n,w}; |P^{(j)}(ξ)|_{w} ≤ X_{j} for j = 0, …, n\}

at the selected place w and

C_{v}(\underline{X}) = \{P ∈ E_{n,v}; ∥P ∥_{v} ≤ 1\}

at the other places v ≠ w. For i = 1, …, n + 1, we denote by

λ_{i}(\underline{X}) = λ_{i}(C(\underline{X}))

the i-th minimum of C(\underline{X}) in E_{n}. By definition, this is the smallest positive real number λ such that λC(\underline{X}) contains i linearly independent elements of E_{n}, where λC(\underline{X}) is the adelic convex body whose component at any Archimedean place v consists of all products λP with P ∈ C_{v}(\underline{X}) and whose component at any ultrametric place v is C_{v}(\underline{X}).

In order to apply the adelic Minkowski’s theorem of [3] in this context, we identify each space E_{n,v} with K_{v}^{n+1} in the natural way described in §2. This identifies ∏_{v} E_{n,v} with (K_{A})^{n+1} where K_{A} denotes the ring of adèles of K, and we use the same Haar measure as in [3] on this space. Explicitly, this means that, for an Archimedean place v of K, we choose the Haar measure on K_{v} to be the Lebesgue measure if K_{v} = R and twice the Lebesgue measure if K_{v} = C. For an ultrametric place v, we normalize the measure so that the ring of integers O_{v} of K_{v} has volume |D_{v}|_{v}^{d/2} where D_{v} denotes the local different of K at v. On each factor E_{n,v} ≃ K_{v}^{n+1} we use the product measure, and we take the product of these measures on ∏_{v} E_{n,v}.

Lemma 3.1. There are two constants c_{1} and c_{2} which depend only on K, n and w such that

c_{1}(X_{0} ⋯ X_{n})^{d} ≤ Vol(C(\underline{X})) ≤ c_{2}(X_{0} ⋯ X_{n})^{d}

for any (n + 1)-tuple of positive real numbers \underline{X} = (X_{0}, X_{1}, …, X_{n}).

Proof. Since the linear map from E_{n,w} to itself sending a polynomial P (T) to P (T + ξ) has determinant 1, the volume of C_{w}(\underline{X}) is equal to that of

{P ∈ E_{n,w} ; |P^{(j)}(0)|_{w} ≤ X_{j} for j = 0, …, n} ≃ ∏_{j=0}^{n} {a ∈ K_{w} ; |j!a|_{w} ≤ X_{j}}

Approximation by conjugate integers

which in turn is bounded above and below by c′_{w}(X_{0} ⋯ X_{n})^{d} and c′′_{w}(X_{0} ⋯ X_{n})^{d} respectively, for some positive constants c′_{w} and c′′_{w} depending only on K, n and w. For the other places v ≠ w, the volume of C_{v}(\underline{X}) is a positive constant c_{v} also depending only on K, n and v, with c_{v} = 1 for almost all places. The conclusion follows.

Lemma 3.2. Let κ be a positive real number and let \underline{X} = (X_{0}, X_{1}, …, X_{n}) be a (n + 1)- tuple of positive real numbers with

X_{0} ⋯ X_{n} ≥ c_{1}^{−1/d}(2/κ)^{n+1}  (3.1)

where c_{1} is the constant of Lemma 3.1. Then, κC(\underline{X}) contains a non-zero element of E_{n}.

Proof. According to Theorem 3 of [3], we have

(λ_{1}(\underline{X}) ⋯ λ_{n+1}(\underline{X}))^{d}Vol(C(\underline{X})) ≤ 2^{d(n+1)}.  (3.2)

Since λ_{1}(\underline{X}) ≤ ⋯ ≤ λ_{n+1}(\underline{X}) and since Vol(C(\underline{X})) ≥ (2/κ)^{d(n+1)} by Lemma 3.1 and condition (3.1), this implies λ_{1}(\underline{X}) ≤ κ, as required.

Note that, for any integer t with 1 ≤ t ≤ n and any real number X ≥ 1, the condition (3.1) is satisfied with

κ = 1, X_{0} = ⋯ = X_{n−t} = cX^{−t/(n+1−t)} and X_{n−t+1} = … = X_{n} = X

for an appropriate constant c. Then, the corresponding convex body C(\underline{X}) contains a nonzero element of E_{n}. In other words, for any integer t with 1 ≤ t ≤ n and any real number X ≥ 1, there exists a nonzero polynomial Q ∈ K[T] of degree at most n which satisfies ∥Q∥_{v} ≤ 1 at each place v of K distinct from w and also

max_{0≤j≤n−t} |Q^{(j)}(ξ)|_{w} ≤ cX^{−t/(n+1−t)} and max_{n−t<j≤n} |Q^{(j)}(ξ)|_{w} ≤ X.

This justifies the remark made after the statement of Theorem B, on comparing with the conditions (1.2) of this theorem.

Our last objective of this section is to relate the successive minima of a convex C(X_{0}, …, X_{n}) with those of C(X_{n}^{−1}, …, X_{0}^{−1}). We achieve this, following ideas that go back to Mahler (see [11] and §VIII.5 of [6]), by showing that these convex bodies are almost

Approximation by conjugate integers

reciprocal with respect to some bilinear form g on E_{n}. This will require two lemmas. The first one defines this bilinear form g and shows a translation invariance property of it.

Lemma 3.3. Let g: E_{n} × E_{n} → K be the K-bilinear form given by the formula

g(P, Q) = ∑_{j=0}^{n}(−1)^{j}P^{(j)}(0)Q^{(n−j)}(0)

for any choice of polynomials P, Q ∈ E_{n}. For each place v of K, denote by g_{v}: E_{n,v}×E_{n,v} → K_{v} the K_{v}-bilinear form which extends g. Then, for any polynomials P, Q ∈ E_{n,w}, we have

g_{w}(P, Q) = ∑_{j=0}^{n}(−1)^{j}P^{(j)}(ξ)Q^{(n−j)}(ξ).

Proof. For fixed P, Q ∈ E_{n,w}, the polynomial

A(T) = ∑_{j=0}^{n}(−1)^{j}P^{(j)}(T)Q^{(n−j)}(T)

has derivative

A′(T) = ∑_{j=0}^{n−1}(−1)^{j}P^{(j+1)}(T)Q^{(n−j)}(T) + ∑_{j=1}^{n}(−1)^{j}P^{(j)}(T)Q^{(n−j+1)}(T) = 0.

So A(T) is a constant. This implies A(ξ) = A(0) = g_{w}(P, Q).

Using this we get the following estimate.

Lemma 3.4. Let \underline{X} = (X_{0}, X_{1}, …, X_{n}) and \underline{Y} = (Y_{0}, Y_{1}, …, Y_{n}) be (n + 1)-tuples of positive real numbers. Suppose that, for each place v of K, we are given polynomials P_{v} ∈ C_{v}(\underline{X}) and Q_{v} ∈ C_{v}(\underline{Y}). Then, with the notation of Lemma 3.3, we have

∏_{v} |g_{v}(P_{v}, Q_{v})|_{v} ≤ (n + 1)! max_{0≤j≤n} X_{j}Y_{n−j}.

Proof. For any place v of K with v ≠ w, we have, if v is Archimedean,

|g_{v}(P_{v}, Q_{v})|_{v} ≤ (n + 1)^{d_{v}/d} max_{0≤j≤n} |P_{v}^{(j)}(0)|_{v}|Q_{v}^{(n−j)}(0)|_{v}
≤ ((n + 1)!)^{d_{v}/d}∥P_{v}∥_{v}∥Q_{v}∥_{v}
≤ ((n + 1)!)^{d_{v}/d},

Approximation by conjugate integers

and, if v is ultrametric,

|g_{v}(P_{v}, Q_{v})|_{v} ≤ max_{0≤j≤n} |P_{v}^{(j)}(0)|_{v}|Q_{v}^{(n−j)}(0)|_{v} ≤ ∥P_{v}∥_{v}∥Q_{v}∥_{v} ≤ 1.

Similarly, if w is Archimedean, the formula of Lemma 3.3 leads to

\begin{aligned}|g_{w}(P_{w}, Q_{w})|_{w} & ≤ (n + 1)^{d_{w}/d} max_{0≤j≤n} |P_{w}^{(j)}(ξ)|_{w}|Q_{w}^{(n−j)}(ξ)|_{w} \\ & ≤ (n + 1)^{d_{w}/d} max_{0≤j≤n} X_{j}Y_{n−j} \end{aligned}

while, if w is non-Archimedean, it gives

|g_{w}(P_{w}, Q_{w})|_{w} ≤ max_{0≤j≤n} |P_{w}^{(j)}(ξ)|_{w}|Q_{w}^{(n−j)}(ξ)|_{w} ≤ max_{0≤j≤n} X_{j}Y_{n−j}.

The conclusion follows.

Proposition 3.5. Let \underline{X} = (X_{0}, X_{1}, …, X_{n}) be an (n+1)-tuple of positive real numbers. Define \underline{Y} = (Y_{0}, …, Y_{n}) where Y_{i} = X_{n−i}^{−1} for i = 0, …, n. Then the products

λ_{i}(\underline{X}) λ_{n−i+2}(\underline{Y}), (1 ≤ i ≤ n + 1),

are bounded below and above by positive constants that depend only on K, n and w.

Proof. Fix an integer i with 1 ≤ i ≤ n + 1. Put λ = λ_{i}(\underline{X}) et μ = λ_{n−i+2}(\underline{Y}). By definition of the successive minima of a convex body, the polynomials of K[T] contained in λC(\underline{X}) generate a subspace U of E_{n} of dimension ≥ i while those contained in μC(\underline{Y}) generate a subspace V of E_{n} of dimension ≥ n − i + 2. Since the sum of these dimensions is strictly greater than that of E_{n} and since the bilinear form g of Lemma 3.3 is nondegenerate, it follows that U and V are not orthogonal with respect to g. Thus, there exist non-zero polynomials P ∈ λC(\underline{X}) and Q ∈ μC(\underline{Y}) which belong to E_{n} and satisfy g(P, Q) ≠ 0. For any Archimedean place v of K, we view λ and μ as elements of K_{v} under the natural embedding of R in K_{v} and define P_{v} = λ^{−1}P and Q_{v} = μ^{−1}Q. For all the other places of K, we put P_{v} = P and Q_{v} = Q. Then, we have P_{v} ∈ C_{v}(\underline{X}) and Q_{v} ∈ C_{v}(\underline{Y}) for all places v of K, and applying Lemma 3.4 we get

∏_{v |̸ ∞} |g(P, Q)|_{v} ∏_{v|∞} |g_{v}(λ^{−1}P, μ^{−1}Q)|_{v} ≤ (n + 1)!.

Approximation by conjugate integers

On noting that, for any Archimedean place v of K, the real numbers λ and μ viewed as elements of K_{v} satisfy |λ|_{v} = λ^{d_{v}/d} and |μ|_{v} = μ^{d_{v}/d}, we find that the left hand side of the above inequality is

∏_{v|∞}(λμ)^{−d_{v}/d} ∏_{v} |g(P, Q)|_{v} = (λμ)^{−1},

by virtue of the product formula applied to the non-zero element g(P, Q) of K. This shows λμ ≥ ((n+1)!)^{−1}, and so all products λ_{i}(\underline{X})λ_{n−i+2}(\underline{Y}) are bounded below by ((n+1)!)^{−1}, for i = 1, …, n + 1.

On the other hand, applying Theorem 3 of [3] to both C(\underline{X}) and C(\underline{Y}) (see (3.2) above), we find

∏_{i=1}^{n+1} (λ_{i}(\underline{X})λ_{n−i+2}(\underline{Y})) = (∏_{i=1}^{n+1} λ_{i}(\underline{X})) (∏_{i=1}^{n+1} λ_{i}(\underline{Y}))
≤ 4^{n+1}Vol(C(\underline{X}))^{−1/d} Vol(C(\underline{Y}))^{−1/d}
≤ 4^{n+1}c_{1}^{−2/d}

where c_{1} is the constant of Lemma 3.1. Thus the products λ_{i}(\underline{X})λ_{n−i+2}(\underline{Y}) are also bounded above by 4^{n+1}c_{1}^{−2/d}((n + 1)!)^{n}.

4. A version of Gel’fond’s criterion.

Let n, w and ξ be as in the preceding section. We prove below a version of Gel’fond’s lemma which contains Theorem 2b of [4] and which we will need to conclude the proof of Theorem B. It applies as well to the situation of Lemma 12 in §10 of [4]. We first need a lemma.

Lemma 4.1. Let P, Q ∈ K[T] be non-zero polynomials of degree at most n, and let G ∈ K[T] be a gcd of P and Q. Then, we have

\frac{|G(ξ)|_{w}}{∥G∥_{w}} ≤ c_{3} max \{\frac{|P (ξ)|_{w}}{∥P ∥_{w}} , \frac{|Q(ξ)|_{w}}{∥Q∥_{w}} \} (\frac{H(P)}{H(G)})^{deg(Q/G)} (\frac{H(Q)}{H(G)})^{deg(P/G)}

where c_{3} = c_{3}(n) is a constant depending only on n.

Proof. Put A = P/G and B = Q/G. For any place v of K, the usual representation of the resultant Res(A, B) of A and B as a Sylvester’s determinant leads to the estimate

|Res(A, B)|_{v} ≤ c_{v}∥A∥_{v}^{degB}∥B∥_{v}^{degA}

Approximation by conjugate integers

where c_{v} = 1 if v |̸ ∞ and c_{v} = ((2n)!)^{d_{v}/d} if v|∞. Arguing as D. Brownawell in the proof of Lemma 1 of [1], we also find

|Res(A, B)|_{w} ≤ c_{w} max \{\frac{|A(ξ)|_{w}}{∥A∥_{w}} , \frac{|B(ξ)|_{w}}{∥B∥_{w}} \} ∥A∥_{w}^{degB}∥B∥_{w}^{degA}

with the same value of c_{w} as above. Since A and B are relatively prime, their resultant is non-zero and so the product formula gives

1 = ∏_{v} |Res(A, B)|_{v} ≤ (2n)! max \{\frac{|A(ξ)|_{w}}{∥A∥_{w}} , \frac{|B(ξ)|_{w}}{∥B∥_{w}} \} H(A)^{degB}H(B)^{degA}.

Multiplying both terms of this inequality by |G(ξ)|_{w}/∥G∥_{w} and using the estimates H(A) ≤ e^{n}H(P)/H(G) and ∥A∥_{w} ≥ (n + 1)^{−1}∥P ∥_{w}/∥G∥_{w} as well as the corresponding estimates for B, we get the desired result with c_{3} = (n + 1)(2n)!e^{2n^{2}}.

Theorem 4.2. Suppose that, for any sufficiently large real number X, there is a non-zero polynomial P = P_{X} ∈ K[T] of degree ≤ n and height ≤ X such that

\frac{|P (ξ)|_{w}}{∥P ∥_{w}} < c_{4}^{−1}H(P)^{−n}X^{−deg(P)}

where c_{4} = e^{2n^{2}}c_{3}. Then, ξ is algebraic over K of degree ≤ n and the above polynomials vanish at ξ for any sufficiently large X.

Proof. Suppose, on the contrary, that there exist arbitrarily large values of X for which P_{X}(ξ) ≠ 0. We will show by induction that, for each integer m = 1, …, n, there exist polynomials Q ∈ K[T] of degree ≤ m which have arbitrarily large height and satisfy

0 < \frac{|Q(ξ)|_{w}}{∥Q∥_{w}} < c_{4}^{−1}H(Q)^{−n−deg(Q)}.  (4.1)

For m = n, the polynomials P_{X} for which P_{X}(ξ) ≠ 0 satisfy the above inequality and, since |P_{X}(ξ)|_{w}/∥P_{X}∥_{w} tends to 0 as X → ∞, their heights tend to infinity with X. So, the above condition is satisfied when m = n. Suppose more generally that it is verified for an integer m with 1 ≤ m ≤ n. Choose a polynomial Q of degree ≤ m which satisfy (4.1) and whose height is sufficiently large so that, putting X = e^{−n}H(Q), the polynomial P = P_{X} is defined. Then Q does not divide P since H(P) ≤ e^{−n}H(Q). So, if G denotes a

Approximation by conjugate integers

gcd of P and Q, then G has degree ≤ m − 1. Writing a = deg(P/G) and b = deg(Q/G), and applying Lemma 4.1, we get

\frac{|G(ξ)|_{w}}{∥G∥_{w}} < c_{3}c_{4}^{−1} max \{H(P)^{−n}X^{−degP} , H(Q)^{−n−degQ}\} (\frac{H(P)}{H(G)})^{b} (\frac{H(Q)}{H(G)})^{a}
= e^{−2n^{2}}H(G)^{−n}X^{−deg(G)}

max \{(\frac{H(G)}{H(P)})^{n−b} (\frac{H(Q)}{XH(G)})^{a} , (\frac{H(G)}{H(Q)})^{n−a} (\frac{H(P)}{XH(G)})^{b} e^{−ndeg(Q)}\} .

Using H(G)/H(P) ≤ e^{n}, H(G)/H(Q) ≤ e^{n}, H(P) ≤ X and again H(Q) = e^{n}X, this gives

\frac{|G(ξ)|_{w}}{∥G∥_{w}} < H(G)^{−n}X^{−deg(G)} max \{H(G)^{−a}, H(G)^{−b}\} .  (4.2)

Since Q(ξ) ≠ 0, we also have G(ξ) ≠ 0. If G were a constant, we would have H(G) = 1 and |G(ξ)|_{w}/∥G∥_{w} = 1 and the inequality (4.2) would not hold. So, the degree of G is ≥ 1 and (4.2) implies that |G(ξ)|_{w}/∥G∥_{w} can be made arbitrarily small by making X arbitrarily large, that is by choosing Q of sufficiently large height. Thus the height of G also can be made arbitrarily large. In particular, we may assume that it is ≥ c_{4}e^{n^{2}}. If a = deg(P/G) ≥ 1, the inequality (4.2) then gives

\frac{|G(ξ)|_{w}}{∥G∥_{w}} < H(G)^{−n−1}X^{−deg(G)} ≤ c_{4}^{−1}H(G)^{−n−deg(G)}

using b = deg(Q/G) ≥ 1 and 1 ≤ H(G) ≤ e^{n}H(P) ≤ e^{n}X. If deg(P/G) = 0, we also find

\frac{|G(ξ)|_{w}}{∥G∥_{w}} = \frac{|P (ξ)|_{w}}{∥P ∥_{w}} < c_{4}^{−1}H(P)^{−n−deg(P)} = c_{4}^{−1}H(G)^{−n−deg(G)}.

If m ≥ 2, this shows that the condition (4.1) is satisfied by polynomials of degree ≤ m − 1 and arbitrarily large height. Thus, our claim is verified for m = 1, …, n. For m = 1, the above argument leads to a contradiction since it constructs non-constant polynomials of degree 0.

5. A height estimate.

Here we establish a height estimate which, in our application, will play the role of Theorem 3 of [4]. Again, we start with a lemma.

Approximation by conjugate integers

Lemma 5.1. Let ℓ ≥ 0 be an integer and let x_{0}, …, x_{ℓ} be indeterminates. For any integer k ≥ 1, the set Z[x_{0}, …, x_{ℓ}]_{k} of homogeneous polynomials of Z[x_{0}, …, x_{ℓ}] of degree k is generated, as a Z-module, by the minors of order k of the k × (k + ℓ) matrix

R(k, ℓ) = ( \begin{matrix}x_{0} & x_{1} & … & x_{ℓ} & 0 & … & 0 \\ 0 & x_{0} & x_{1} & … & x_{ℓ} & ... & ⋮ \\ ⋮ & ... & ... & ... & & ... & 0 \\ 0 & … & 0 & x_{0} & x_{1} & … & x_{ℓ} \end{matrix} )  \} k rows.

Proof. We proceed by induction on k + ℓ. If k = 1 or ℓ = 0 the result is clear. Assume k ≥ 2 and ℓ ≥ 1 and that the result is true for a smaller number of rows or a smaller number of indeterminates. Denote by M the subgroup of Z[x_{0}, …, x_{ℓ}]_{k} generated by the minors of order k of the matrix R(k, ℓ). The ring homomorphism φ from Z[x_{0}, …, x_{ℓ}] to Z[x_{0}, …, x_{ℓ−1}] sending x_{ℓ} to 0 and all other indeterminates to themselves maps M onto the subgroup of Z[x_{0}, …, x_{ℓ−1}]_{k} generated by the minors of order k of R(k, ℓ − 1). Thus, by the induction hypothesis, we have

φ(M) = Z[x_{0}, …, x_{ℓ−1}]_{k}.

On the other hand, the determinants of the k × k sub-matrices which contain the last column of R(k, ℓ) are the products x_{ℓ}d where d is a minor of order k − 1 of R(k − 1, ℓ). Thus, by the induction hypothesis, we also have

M ⊇ x_{ℓ}Z[x_{0}, …, x_{ℓ}]_{k−1} = Z[x_{0}, …, x_{ℓ}]_{k} ∩ ker(φ).

These properties imply M = Z[x_{0}, …, x_{ℓ}]_{k}.

Proposition 5.2. Let k and ℓ be integers with k ≥ 1 and ℓ ≥ 0. For any P ∈ E_{ℓ}, we have

c^{−1}H(P)^{k} ≤ H(P · E_{k−1}) ≤ cH(P)^{k}

where c is a positive constant depending only on k and ℓ.

Proof. Write P = a_{0} + a_{1}T + ⋯ + a_{ℓ}T^{ℓ}. Then the height of P · E_{k−1} is simply the height of the matrix with k rows

R = ( \begin{matrix}a_{0} & a_{1} & … & a_{ℓ} & & 0 \\ & ... & ... & & ... \\ 0 & & a_{0} & a_{1} & … & a_{ℓ} \end{matrix} ) .

Approximation by conjugate integers

By virtue of the preceding lemma, every monomial of degree k in a_{0}, …, a_{ℓ} can be expressed as a linear combination of the minors of order k of this matrix with integral coefficients that do not depend on P. Conversely, the minors of order k of R can be written as linear combinations of monomials of degree k in a_{0}, …, a_{ℓ} with integral coefficients that do not depend on P. Thus, for each place v of K, we have

c_{v}^{−1}∥R∥_{v} ≤ ∥P ∥_{v}^{k} ≤ c_{v}∥R∥_{v}

for some constant c_{v} ≥ 1 independent of P, with c_{v} = 1 when v is not Archimedean. The conclusion follows with c = ∏_{v|∞} c_{v}.

6. Construction of a polynomial.

Let n, w and ξ be as in §3. We fix a non-decreasing sequence of positive real numbers X_{0} ≤ ⋯ ≤ X_{n} and assume that the corresponding convex body C(X_{0}, …, X_{n}) contains a non-zero polynomial Q ∈ K[T]. Let

V = {P ∈ E_{n}; g(P, Q) = 0}

where g: E_{n} × E_{n} → K is the K-bilinear form of Lemma 3.3. For each integer ℓ with 0 ≤ ℓ ≤ n, we define a K-bilinear form B_{ℓ}: E_{ℓ} × E_{n−ℓ} → K by the formula

B_{ℓ}(F, G) = g(F G, Q)

for F ∈ E_{ℓ} and G ∈ E_{n−ℓ}. Its right kernel is

V_{ℓ} = {G ∈ E_{n−ℓ}; G · E_{ℓ} ⊆ V }

We also denote by B_{ℓ,w}: E_{ℓ,w} × E_{n−ℓ,w} → K_{w} the K_{w}-bilinear form which extends B_{ℓ}. Finally, we put

y_{i} = (−1)^{i}i!Q^{(n−i)}(0) and z_{i} = (−1)^{i}i!Q^{(n−i)}(ξ), (0 ≤ i ≤ n),

and, for each integer ℓ = 0, 1, …, n, we define

M_{ℓ} = ( \begin{matrix}y_{0} & y_{1} & … & y_{n−ℓ} \\ y_{1} & y_{2} & … & y_{n−ℓ+1} \\ ⋮ & ⋮ & & ⋮ \\ y_{ℓ} & y_{ℓ+1} & … & y_{n} \end{matrix} ) and N_{ℓ} = ( \begin{matrix}z_{0} & z_{1} & … & z_{n−ℓ} \\ z_{1} & z_{2} & … & z_{n−ℓ+1} \\ ⋮ & ⋮ & & ⋮ \\ z_{ℓ} & z_{ℓ+1} & … & z_{n} \end{matrix} ) .

Approximation by conjugate integers

With this notation, we will prove below a series of lemmas leading, under some condition on X_{0}, …, X_{n}, to the construction of a polynomial P ∈ K[T] with several properties. The method overall follows that of Davenport and Schmidt in §§7–9 of [4]. The first lemma is the following observation:

Lemma 6.1. Fix an integer ℓ with 0 ≤ ℓ ≤ n. Then,

(i) M_{ℓ} is the matrix of B_{ℓ} relative to the bases {1, T, …, T^{ℓ}} of E_{ℓ} and {1, T, …, T^{n−ℓ}} of E_{n−ℓ};

(ii) N_{ℓ} is the matrix of B_{ℓ,w} relative to the bases {1, T − ξ, …, (T − ξ)^{ℓ}} of E_{ℓ,w} and {1, T − ξ, …, (T − ξ)^{n−ℓ}} of E_{n−ℓ,w}.

Proof. This follows upon noting that, for i = 0, …, ℓ and j = 0, …, n − ℓ, we have

B_{ℓ}(T^{i}, T^{j}) = g(T^{i+j}, Q(T)) = y_{i+j}

and, by Lemma 3.3,

B_{ℓ,w}((T − ξ)^{i}, (T − ξ)^{j}) = g_{w}((T − ξ)^{i+j}, Q(T)) = z_{i+j}.

In particular, this result implies that M_{ℓ} and N_{ℓ} have the same rank for any value of ℓ. Note that the number of rows of these matrices is less than or equal to their number of columns if and only if ℓ ≤ n/2. Under this hypothesis, we have the following estimates:

Lemma 6.2. There are constants c_{5}, c_{6}, c_{7} ≥ 1 depending only on K, n, w and ξ such that, for any integer ℓ with 0 ≤ ℓ ≤ n/2, we have

(i) ∥N_{ℓ}∥_{w} ≤ c_{5}X_{n−ℓ} ⋯ X_{n}, (ii) c_{6}^{−1}∥N_{ℓ}∥_{w} ≤ ∥M_{ℓ}∥_{w} ≤ c_{6}∥N_{ℓ}∥_{w}, (iii) H(M_{ℓ}) ≤ c_{7}∥N_{ℓ}∥_{w} when M_{ℓ} has rank ℓ + 1.

Proof. (i) The upper bound on ∥N_{ℓ}∥_{w} follows from the fact that, for i = 1, …, ℓ + 1, all the elements of the i-th row of N_{ℓ} have their absolute value bounded above by a constant times X_{n+1−i}.

(ii) By Lemma 6.1, M_{ℓ} and N_{ℓ} are matrices of B_{ℓ,w} corresponding to different choices of bases for E_{ℓ,w} and E_{n−ℓ,w}. Accordingly, we have

M_{ℓ} = ^{t}UN_{ℓ}V,

Approximation by conjugate integers

where U and V are matrices of change of bases which depend only on ξ, ℓ and n. Since U and V are invertible, this implies that any minor of order ℓ + 1 of M_{ℓ} (resp. N_{ℓ}) can be expressed as a linear combination of the minors of order ℓ + 1 of N_{ℓ} (resp. M_{ℓ}) with coefficients that are independent of Q, and the second assertion follows.

(iii) At any place v of K with v ≠ w, the elements of M_{ℓ} have their absolute value bounded above by a constant which depends only on n and which can be taken to be 1 when v |̸ ∞. So, the same is true of ∥M_{ℓ}∥_{v}. The height of M_{ℓ} is thus bounded above by a constant times ∥M_{ℓ}∥_{w} or, according to (ii), by a constant times ∥N_{ℓ}∥_{w}.

Lemma 6.3. For any integer ℓ with 0 ≤ ℓ ≤ n, we have

dim V_{ℓ} = n − ℓ + 1 − rank(M_{ℓ}).

When M_{ℓ} has rank ℓ + 1, we also have H(V_{ℓ}) = H(M_{ℓ}).

Proof. A polynomial P = a_{0} + a_{1}T + ⋯ + a_{n−ℓ}T^{n−ℓ} of E_{n−ℓ} belongs to V_{ℓ} if and only if for i = 0, …, ℓ, it satisfies

0 = g(T^{i}P (T), Q(T)) = B_{ℓ}(T^{i}, P (T)) = ∑_{j=0}^{n−ℓ} y_{i+j}a_{j}.

Thus, identifying E_{n−ℓ} with K^{n−ℓ+1} in the usual way, the subspace V_{ℓ} of E_{n−ℓ} is identified with the solution space of the homogeneous system associated to M_{ℓ}. This proves the formula for dim V_{ℓ}. Moreover, if M_{ℓ} has rank ℓ+1, then, according to the duality principle mentioned in §2, we have H(V_{ℓ}) = H(M_{ℓ}).

Lemma 6.4. Suppose that there exists an integer h with 1 ≤ h ≤ n/2 such that M_{h−1} has rank h and M_{h} has rank ≤ h. Then, V_{n−h} contains a non-zero element P. Such a polynomial P has degree ≤ h and satisfies

P · E_{n−2h+1} = V_{h−1}.

In particular, P divides any polynomial of V_{h−1}.

Proof. Since M_{n−h} is the transpose of M_{h}, the two matrices have the same rank. By Lemma 6.3, this gives

dim V_{n−h} = (h + 1) − rank(M_{n−h}) ≥ 1.

Approximation by conjugate integers

So, V_{n−h} contains a non-zero element P. Using Lemma 6.3, we also find

dim V_{h−1} = (n − h + 2) − rank(M_{h−1}) = n − 2h + 2.

Since V_{h−1} contains P ·E_{n−2h+1}, and since the latter subspace of E_{n−h+1} also has dimension n − 2h + 2, this inclusion is an equality. Lemma 6.5. Let ℓ and t be integers with 0 ≤ ℓ < n/2 and 1 ≤ t ≤ n − 2ℓ. Suppose that N_{ℓ} has rank ℓ + 1 and that there exists a non-zero polynomial P ∈ K[T] such that P · E_{t−1} ⊆ V_{ℓ} . Then, we have

(\frac{|P (ξ)|_{w}}{∥P ∥_{w}} )^{t} ≪ \frac{X_{n−t−ℓ} ⋯ X_{n−t}}{∥N_{ℓ}∥_{w}} .

Proof. Denote by z_{0}, …, z_{n−ℓ} the columns of N_{ℓ} and, for each integer s with 1 ≤ s ≤ t+1, denote by N_{ℓ}^{(s)} the sub-matrix of N_{ℓ} consisting of the columns z_{s−1}, …, z_{n−ℓ}. Observe that, since t ≤ n − 2ℓ, these matrices all have at least ℓ + 1 columns. Write

P = b_{0} + b_{1}(T − ξ) + ⋯ + b_{h}(T − ξ)^{h}

where h is the degree of P. For any integer s as above, we have (T − ξ)^{s−1}P (T) ∈ V_{ℓ} and so, for i = 0, …, ℓ, we find

0 = B_{ℓ,w}((T − ξ)^{i}, (T − ξ)^{s−1}P (T)) = ∑_{j=0}^{h} z_{i+s−1+j}b_{j}.

This means that the columns of N_{ℓ} satisfy the recurrence relation

b_{0}z_{s−1} + b_{1}z_{s} + ⋯ + b_{h}z_{s−1+h} = 0, (1 ≤ s ≤ t).

Now, fix an integer s with 1 ≤ s ≤ t and choose indices j_{0}, j_{1}, …, j_{ℓ} with s − 1 ≤ j_{0} < j_{1} < ⋯ < j_{ℓ} ≤ n − ℓ such that

∥N_{ℓ}^{(s)}∥_{w} = | det(z_{j_{0}}, …, z_{j_{ℓ}})|_{w}.  (6.1)

If j_{0} = s − 1, we find, using the recurrence relation,

|P (ξ)|_{w}∥N_{ℓ}^{(s)}∥_{w} = | det(b_{0}z_{s−1}, z_{j_{1}}, …, z_{j_{ℓ}})|_{w}

= | det ( − ∑_{j=1}^{h} b_{j}z_{s−1+j}, z_{j_{1}}, …, z_{j_{ℓ}})|_{w}

= | ∑_{j=1}^{h} b_{j} det(z_{s−1+j}, z_{j_{1}}, …, z_{j_{ℓ}})|_{w}

≤ c∥P ∥_{w}∥N_{ℓ}^{(s+1)}∥_{w}

Approximation by conjugate integers

for some positive constant c depending only on n and |ξ|_{w}. If j_{0} ≥ s, this is still true because (6.1) then implies ∥N_{ℓ}^{(s)}∥_{w} ≤ ∥N_{ℓ}^{(s+1)}∥_{w}. Since ∥N_{ℓ}^{(1)}∥_{w} = ∥N_{ℓ}∥_{w} ≠ 0, this inequality implies by induction on s that we have ∥N_{ℓ}^{(s)}∥_{w} ≠ 0 for s = 1, …, t + 1, and therefore we can write

\frac{|P (ξ)|_{w}}{∥P ∥_{w}} ≤ c\frac{∥N_{ℓ}^{(s+1)}∥_{w}}{∥N_{ℓ}^{(s)}∥_{w}} , (1 ≤ s ≤ t).

Multiplying term by term these inequalities, we get

(\frac{|P (ξ)|_{w}}{∥P ∥_{w}} )^{t} ≤ c^{t} \frac{∥N_{ℓ}^{(t+1)}∥_{w}}{∥N_{ℓ}∥_{w}}

and the conclusion follows upon noting that, for i = 1, …, ℓ + 1, the i-th row of N_{ℓ}^{(t+1)} has norm ≪ X_{n−t−i+1} and thus ∥N_{ℓ}^{(t+1)}∥_{w} ≪ X_{n−t−ℓ} ⋯ X_{n−t}.

Proposition 6.6. Let k be an integer with 1 ≤ k ≤ n/2. Assume that there is an integer t with 1 ≤ t ≤ n + 2 − 2k such that

X_{0} ≤ ⋯ ≤ X_{n−t} < 1 and 1 ≤ X_{n−t+1} ≤ ⋯ ≤ X_{n}.

Put δ = X_{n−t} and Y = X_{n−t+1} ⋯ X_{n}, and assume moreover that

Y δ^{k+1−t} < (c_{5}c_{7})^{−1}  (6.2)

where c_{5} and c_{7} are defined in Lemma 6.2. Then there exists an integer h with 1 ≤ h ≤ k and a non-zero polynomial P ∈ K[T] of degree ≤ h and height ≪ δ^{−k/n} which divides any polynomial of V_{h−1} and satisfies

(\frac{|P (ξ)|_{w}}{∥P ∥_{w}} )^{t} ≤ c_{8}δ^{h}H(P)^{−(n+2−2h)},  (6.3)

where c_{8} is a constant depending only on K, n, w and ξ.

Proof. For any integer ℓ for which M_{ℓ} has rank ℓ + 1, we find, using Lemma 6.2,

H(M_{ℓ}) ≤ c_{7}∥N_{ℓ}∥_{w} ≤ c_{5}c_{7}X_{n−ℓ} ⋯ X_{n} ≤ c_{5}c_{7}Y δ^{ℓ+1−t}.  (6.4)

Since we also have H(M_{ℓ}) ≥ 1 for these values of ℓ, the assumption (6.2) implies that M_{k} has rank ≤ k. The rank of M_{0} being 1, we conclude that there exists an integer h with

Approximation by conjugate integers

1 ≤ h ≤ k such that M_{h−1} has rank h and M_{h} has rank ≤ h. Then, according to Lemma 6.4, there exists a non-zero polynomial P ∈ E_{h} such that

P · E_{n−2h+1} = V_{h−1}.

This implies that P divides any polynomial of V_{h−1} and, by Proposition 5.2, that

H(P)^{n+2−2h} ≪ H(V_{h−1}) ≪ H(P)^{n+2−2h}.  (6.5)

Combining Lemma 6.3 with (6.2) and (6.4) (for ℓ = h − 1), we also find

H(V_{h−1}) = H(M_{h−1}) ≤ c_{5}c_{7}Y δ^{h−t} < δ^{−(k+1−h)}.  (6.6)

Note that, since k ≤ n/2, the ratio (k + 1 − h)/(n + 2 − 2h) is a decreasing function of h in the range 1 ≤ h ≤ k. So, it is bounded above by k/n. Combining this observation with the above estimates (6.5) and (6.6), we get

H(P) ≪ δ^{−(k+1−h)/(n+2−2h)} ≪ δ^{−k/n}.  (6.7)

Since t ≤ n + 2 − 2k, we have P · E_{t−1} ⊆ P · E_{n−2h+1} ⊆ V_{h−1} and applying Lemma 6.5 with ℓ = h − 1 gives

(\frac{|P (ξ)|_{w}}{∥P ∥_{w}} )^{t} ≪ \frac{δ^{h}}{∥N_{h−1}∥_{w}} .

Moreover, Lemma 6.2 (iii), Lemma 6.3 and (6.5) provide

∥N_{h−1}∥_{w} ≥ c_{7}^{−1}H(M_{h−1}) = c_{7}^{−1}H(V_{h−1}) ≫ H(P)^{n+2−2h},

and the conclusion follows.

In our application, we will simply need the following consequence of this proposition.

Corollary 6.7. Assume that all the hypotheses of Proposition 6.6 are satisfied and that we have δ < c_{8}^{−1}. Then there exists an irreducible polynomial P ∈ K[T] which divides any polynomial of V_{k−1} and satisfies

(\frac{|P (ξ)|_{w}}{∥P ∥_{w}} )^{t} ≤ c_{9}δ^{degP} H(P)^{−(n+2−2k)}  (6.8)

Approximation by conjugate integers

where c_{9} = max{1, e^{n^{2}}c_{8}}.

Proof. Let h and P be as in the conclusion of the Proposition. Since H(P) ≥ 1, the right hand side of (6.3) is bounded above by c_{8}δ^{h} ≤ c_{8}δ < 1. So P cannot be a constant. Moreover, since deg(P) ≤ h ≤ k, the same inequality (6.3) gives

(\frac{|P (ξ)|_{w}}{∥P ∥_{w}} )^{t} ≤ c_{8}δ^{deg(P)}H(P)^{−(n+2−2k)}.

Write P as a product

P = P_{1} ⋯ P_{s}

of irreducible polynomials of K[T]. Then the above inequality leads to

∏_{i=1}^{s} ((\frac{|P_{i}(ξ)|_{w}}{∥P_{i}∥_{w}} )^{t} δ^{−deg(P_{i})}H(P_{i})^{n+2−2k}) ≤ e^{n^{2}}c_{8} ≤ c_{9}^{s}.

So, a least one factor of the product on the left must be bounded above by c_{9}. The corresponding polynomial P_{i} divides every element of V_{k−1} since it divides P and V_{k−1} is contained in V_{h−1}.

Note that this statement provides no upper bound on the degree and height of P. We will get such upper bounds by an indirect argument, using the construction of an auxiliary polynomial in the next section.

7. Degree and height estimates.

The notation is as in the preceding section. We assume that the adelic convex body C(X_{0}, …, X_{n}) contains a non-zero polynomial Q of K[T] for some non-decreasing sequence of positive real numbers X_{0} ≤ ⋯ ≤ X_{n}, and we define corresponding subspaces V_{ℓ} of E_{n−ℓ} for ℓ = 0, …, n as in §6.

Lemma 7.1. Put c = ((n + 1)!)^{−2}. Then, for any integer ℓ with 0 ≤ ℓ ≤ n we have

C(cX_{n}^{−1}, …, cX_{ℓ}^{−1}) ∩ E_{n−ℓ} ⊆ V_{ℓ}.  (7.1)

Proof. Let ℓ be an integer with 0 ≤ ℓ ≤ n, and let G be an element of the left hand side of (7.1). We need to show that g(T^{m}G, Q) = 0 for m = 0, …, ℓ. To this end, we proceed by induction. We fix an integer m with 0 ≤ m ≤ ℓ and assume, when m ≥ 1, that we have

Approximation by conjugate integers

g(T^{j}G, Q) = 0 for j = 0, …, m − 1. Let P = T^{m}G(T). We define P_{w} = (T − ξ)^{m}G(T) and Q_{w} = Q and, for the other places v ≠ w of K, we put P_{v} = P and Q_{v} = Q. These polynomials satisfy P_{v} ∈ C_{v}(\underline{Y}) and Q_{v} ∈ C_{v}(\underline{X}) for each place v of K, where \underline{Y} = (Y_{0}, …, Y_{n}) denotes the (n + 1)-tuple of positive real numbers given by Y_{i} = n!cX_{n−i}^{−1} for i = 0, …, n. Moreover, the hypotheses imply

g_{w}(P_{w}, Q_{w}) = g(P, Q)

since the difference P_{w} − P can be written as a linear combination of G, …, T^{m−1}G with coefficients in K_{w} in the case m ≥ 1 and is zero when m = 0. Using Lemma 3.4, we therefore get

∏_{v} |g(P, Q)|_{v} = ∏_{v} |g_{v}(P_{v}, Q_{v})|_{v} ≤ (n + 1)!n!c < 1.

By the product formula, this implies g(P, Q) = 0.

Proposition 7.2. There is a constant c_{10} > 0 which depends only on K, n, w and ξ and has the following property. Suppose that ℓ and u are non-negative integers with ℓ + u < n, such that

X_{n}^{u+1}X_{n−1} ⋯ X_{ℓ+u} ≤ c_{10}.  (7.2)

Then, there is a non-zero polynomial G of K[T] of degree ≤ n − ℓ and height ≪ X_{ℓ+u}^{−1} such that G^{(i)} ∈ V_{ℓ} for i = 0, …, u.

Proof. Let c be as in Lemma 7.1. Put κ = (n!)^{−1} and define real numbers Y_{0}, …, Y_{n−ℓ} by

Y_{i} = \{ \begin{matrix}cX_{n}^{−1} & for i = 0, …, u, \\ cX_{n−i+u}^{−1} & for i = u + 1, …, n − ℓ. \end{matrix}

Lemma 3.2 shows that the convex κC(Y_{0}, …, Y_{n−ℓ}) contains a non-zero element G of E_{n−ℓ} if the condition (7.2) is satisfied for a sufficiently small constant c_{10} > 0. Such a polynomial has height H(G) ≪ Y_{n−ℓ} ≪ X_{ℓ+u}^{−1}. Moreover, for i = 0, …, u, we find

G^{(i)} ∈ C(Y_{i}, …, Y_{n−ℓ}) ⊆ C(cX_{n}^{−1}, …, cX_{ℓ}^{−1})

and so G^{(i)} ∈ V_{ℓ} by Lemma 7.1.

We will apply this proposition in the following context:

Approximation by conjugate integers

Corollary 7.3. Let ℓ and u be as in Proposition 7.2, and assume that there exists an irreducible polynomial P ∈ K[T] which divides every element of V_{ℓ}. Then, we have

deg(P) ≤ \frac{n − ℓ}{u + 1} and H(P) ≪ X_{ℓ+u}^{−1/(u+1)}.

Proof. The hypotheses imply that P divides all derivatives of the polynomial G of Proposition 7.2, up to order u. So, P^{u+1} divides G and the conclusion follows.

8. Proof of Theorem B.

Let the notation be as in Theorem B and assume that the hypothesis of this theorem holds with a constant c < min{1, (c_{5}c_{7})^{−1}}. Then, for any real number X ≥ 1 the condition (6.2) of Proposition 6.6 is satisfied with

δ = X_{0} = X_{1} = ⋯ = X_{n−t} = cX^{−t/(k+1−t)} and X_{n−t+1} = ⋯ = X_{n} = X.

Moreover, if X is sufficiently large, the hypothesis of Theorem B is that the corresponding convex C(\underline{X}) with \underline{X} = (X_{0}, …, X_{n}) contains a non-zero element Q of E_{n}. Since t ≤ n+2−2k, we may then apply Corollary 6.7. It shows that, if X is sufficiently large so that δ < c_{8}^{−1}, then there is an irreducible polynomial P ∈ K[T] which divides every element of the vector space V_{k−1} attached to Q and satisfies

\frac{|P (ξ)|_{w}}{∥P ∥_{w}} ≪ H(P)^{−(n+2−2k)/t}δ^{deg(P)/t}.

Since c ≤ 1 and n − t ≥ 2t + k − 2, we also find

X_{n}^{2t}X_{n−1} ⋯ X_{2t+k−2} = X^{3t−1}δ^{n−3t−k+3} ≤ X^{3t−1}δ^{3k−3t+3} ≤ X^{−1}.

So, the condition 7.2 of Proposition 7.2 is satisfied with ℓ = k − 1 and u = 2t − 1 provided that X is sufficiently large. Assuming that this is the case, Corollary 7.3 then shows

deg(P) ≤ \frac{n − k + 1}{2t} and H(P) ≤ κδ^{−1/(2t)}

for some constant κ > 0. Putting m = [(n − k + 1)/(2t)] and Y = κδ^{−1/(2t)}, and noting that (n + 2 − 2k)/t ≥ m, we thus have found the existence of a polynomial P ∈ K[T] of degree ≤ m and height ≤ Y such that

\frac{|P (ξ)|_{w}}{∥P ∥_{w}} ≪ H(P)^{−m}Y^{−2deg(P)}.

Since Y is a monotone increasing unbounded continuous function of X, for X ≥ 1, Theorem 4.2 then shows that ξ is algebraic over K of degree ≤ m.

Approximation by conjugate integers

9. Proof of Theorem A. We first generalize the construction of Davenport and Schmidt in §2 of [4].

Lemma 9.1. Let t be an integer with 1 ≤ t ≤ n and let δ, κ and Y be real numbers with 0 < δ < 1 < Y and κ ≥ 1. Assume that \underline{Y} = (Y_{0}, …, Y_{n}) is a (n + 1)-tuple of positive real numbers satisfying

λ_{n+1}(\underline{Y}) ≤ κ and Y_{j} ≤ \{ \begin{matrix}Y δ^{t−j} & for j = 0, …, t − 1, \\ Y & for j = t, …, n. \end{matrix}

Assume moreover that |ξ|_{w} ≤ 1 in the case where w is ultrametric. Then there exists an irreducible monic polynomial P ∈ O_{K}[T] of degree n + 1 and height ≪ κY which admits at least t distinct roots in the closed disk of K_{w} centered at ξ of radius δ.

Proof. Choose elements P_{1}, …, P_{n+1} in E_{n} realizing the successive minima of C(\underline{Y}) in E_{n}. Since λ_{n+1}(\underline{Y}) ≤ κ, these polynomials all belong to κC(\underline{Y}). Moreover, they form a basis of E_{n} over K. We will construct the required polynomial P in the form

P (T) = T^{n+1} + ∑_{i=1}^{n+1} b_{i}P_{i}(T)

for suitable elements b_{1}, …, b_{n+1} of K. Each b_{i} will be obtained as solution of a system of inhomogeneous inequalities using the strong approximation theorem (see Theorem 3, page 440 of [12] or §15 of [7]). According to this result, there is a constant C > 0 depending only on K with the following property. For any finite set S of places of K, any choice of elements β_{v} ∈ K_{v}, (v ∈ S), and any choice of positive real numbers ϵ_{v}, (v ∈ S), with ∏_{v∈S} ϵ_{v} ≥ C, there exists an element b of K, satisfying |b − β_{v}|_{v} ≤ ϵ_{v} for v ∈ S, and |b|_{v} ≤ 1 for v ∉ S.

To ensure that P is irreducible, we proceed essentially as Bugeaud and Teulié in [2]. We choose an ultrametric place v_{0} of K with v_{0} ≠ w, and a uniformising parameter π ∈ K_{v_{0}} at v_{0}, i.e. a generator of the maximal ideal of O_{v_{0}}. We write

π = ∑_{i=1}^{n+1} γ_{i}P_{i}(T)

with γ_{1}, …, γ_{n+1} ∈ K_{v_{0}} and we ask that

|b_{i} − γ_{i}|_{v_{0}} ≤ |π|_{v_{0}}^{2}, (1 ≤ i ≤ n + 1).  (9.1)

Approximation by conjugate integers

Under these conditions, the corresponding polynomial P satisfies

∥P (T) − T^{n+1} − π∥_{v_{0}} = ∥ ∑_{i=1}^{n+1}(b_{i} − γ_{i})P_{i}(T)∥_{v_{0}} ≤ |π|_{v_{0}}^{2}.

Thus, P has integral coefficients at the place v_{0} and, according to Eisenstein’s criterion (see for instance Theorem 24 in § 3, Chapter III of [8]), it is irreducible in K_{v_{0}}[T]. Therefore P is irreducible as an element of K[T].

To ensure that P has t roots close to ξ in K_{w}, we fix a monic polynomial B ∈ K_{w}[T] of degree t with t (simple) distinct roots in the open unit disk D_{w} = {z ∈ K_{w} ; |z|_{w} < 1} of K_{w} and we use the fact that, by explicit forms of the inverse function theorem such as Theorem 4.4.1 in Chapter I of [5], any polynomial S(T) ∈ K_{w}[T] of degree ≤ n + 1 for which ∥S −B∥_{w} is sufficiently small also has t distinct roots in D_{w}. We proceed as follows:

Let ϵ be a positive real number with ϵ ≤ 1 and let δ_{0} = min{δ, ϵ}. If w is ultrametric, lying above an ordinary prime number p, we choose an element r of K_{w} with p^{−1}δ_{0} ≤ |r|_{w} ≤ δ_{0} and put s = r^{t}. If w is Archimedean, we choose r, s ∈ K_{w} with |r|_{w} = δ_{0} and |s|_{w} = κ^{d_{w}/d}ϵ^{−t−2}δ_{0}^{t}Y. In both cases, we define

R(T) = (T − ξ)^{n+1} + sB (\frac{T − ξ}{r} ) ∈ K_{w}[T].

We write this polynomial in the form

R(T) = T^{n+1} + ∑_{i=1}^{n+1} θ_{i}P_{i}(T)

with θ_{1}, …, θ_{n+1} ∈ K_{w} and ask that

|b_{i} − θ_{i}|_{w} ≤ \{ \begin{matrix}ϵ^{−1} & if w is Archimedean, \\ ϵ^{n+1}Y^{−1} & if w is ultrametric. \end{matrix}  (9.2)

The polynomial S = s^{−1}P (rT + ξ) then satisfies

∥S − B∥_{w} = ∥s^{−1}r^{n+1}T^{n+1} + s^{−1} ∑_{i=1}^{n+1}(b_{i} − θ_{i})P_{i}(rT + ξ)∥_{w} ≪ ϵ

using |s|_{w}^{−1}|r|_{w}^{n+1} ≪ δ_{0} ≪ ϵ and noting that, for i = 1, …, n + 1, we have

∥P_{i}(rT + ξ)∥_{w} ≪ \{ \begin{matrix}κ^{d_{w}/d}δ^{t}Y & if w is Archimedean, \\ δ^{t}Y & if w is ultrametric. \end{matrix}

Approximation by conjugate integers

If ϵ is sufficiently small, this implies that S has t roots in the disk D_{w} and therefore, that P has at least t distinct roots in the disk of K_{w} centered at ξ with radius δ.

If w is Archimedean and again if ϵ is small enough, the strong approximation Theorem allows us to require, aside from (9.1) and (9.2), that

|b_{i}|_{v} ≤ 1, (1 ≤ i ≤ n + 1),  (9.3)

for all places v of K distinct from v_{0} and w. Then P has integral coefficients at v for each ultrametric place v of K and therefore it has coefficients in O_{K}. Moreover, as we may take ϵ ≫ 1, we find ∥P ∥_{w} ≪ κ^{d_{w}/d}ϵ^{−t−2}Y ≪ κ^{d_{w}/d}Y. Since ∥P ∥_{v} ≪ κ^{d_{v}/d} for all the other Archimedean places v of K, this implies H(P) ≪ ∏_{v|∞} ∥P ∥_{v} ≪ κY.

If w is ultrametric, we choose an Archimedean place v_{1}. We require that (9.1) and (9.2) hold, that (9.3) holds for all places v of K distinct from v_{0}, v_{1} and w, and that

|b_{i}|_{v_{1}} ≤ ϵ^{−n−2}Y, (1 ≤ i ≤ n + 1).

Again, the strong approximation Theorem shows that these conditions have solutions b_{i} ∈ K for i = 1, …, n+1 provided that ϵ is small enough. Then, the corresponding polynomial P has integral coefficients at v for each ultrametric place v of K with v ≠ w. At the place w, we find

∥P − R∥_{w} = ∥∑_{i=1}^{n+1}(b_{i} − θ_{i})P_{i}(T)∥_{w} ≤ ϵ^{n+1}Y^{−1} max_{1≤i≤n+1} ∥P_{i}∥_{w} ≪ ϵ^{n+1}.

Moreover, R has coefficients in O_{w} since |ξ|_{w} ≤ 1 and ∥B∥_{w} ≤ 1. Thus, P also has coefficients in O_{w} if ϵ is sufficiently small, and then it has coefficients in O_{K}. Since we may take ϵ ≫ 1, this gives ∥P ∥_{v_{1}} ≪ κ^{d_{v_{1}} /d}Y and, since ∥P ∥_{v} ≪ κ^{d_{v}/d} for all Archimedean places v ≠ v_{1}, we find H(P) ≪ ∏_{v|∞} ∥P ∥_{v} ≪ κY.

Proof of Theorem A.

Let t, n and ξ be as in Theorem A, let k = [n/4], and let c be the constant of Theorem B corresponding to these data. Since ξ is not algebraic over K of degree ≤ (n − k + 1)/(2t), Theorem B shows that there are arbitrary large positive real numbers X for which the (n + 1)-tuple \underline{X} = (X_{0}, …, X_{n}) given by

X_{j} = \{ \begin{matrix}cX^{−t/(k+1−t)} & for j = 0, …, n − t, \\ X & for j = n − t + 1, …, n \end{matrix}

Approximation by conjugate integers

satisfies λ_{1}(\underline{X}) > 1. According to Proposition 3.5, this implies that the (n + 1)-tuple \underline{Y} = (Y_{0}, …, Y_{n}) given by

Y_{j} = X_{n−j}^{−1} = \{ \begin{matrix}X^{−1} & for j = 0, …, t − 1, \\ c^{−1}X^{t/(k+1−t)} & for j = t, …, n \end{matrix}

satisfies λ_{n+1}(\underline{Y}) ≤ κ with a constant κ ≥ 1 depending only on K, n and w. Assuming X sufficiently large, we may thus apply Lemma 9.1 with

Y = c^{−1}X^{t/(k+1−t)} and δ = c^{1/t}X^{−(k+1)/(t(k+1−t)}.

It shows the existence of an irreducible monic polynomial P ∈ O_{K}[T] of degree n + 1 and height H(P) ≪ Y which admits at least t distinct roots α_{1}, …, α_{t} ∈ K_{w} with

max_{1≤i≤t} |ξ − α_{i}|_{w} ≪ Y^{−(k+1)/t^{2}}.  (9.4)

In particular, α = α_{1} is an algebraic integer of degree n + 1 over K and, using Proposition 2.5 in Chapter 3 of [9], we find H(α) ≪ H(P)^{d} ≪ Y^{d}. Combining this with (9.4), we obtain that the conjugates α_{1}, …, α_{t} of α over K satisfy

max_{1≤i≤t} |ξ − α_{i}|_{w} ≪ H(α)^{−(k+1)/(dt^{2})} ≪ H(α)^{−(n+1)/(4dt^{2})}.

Moreover, since the right hand side of (9.4) can be made arbitrarily small by choosing X sufficiently large, we produce an infinity of such numbers α by varying X.

10. Remarks on simultaneous approximations.

We first show that, for t ≥ 2, when K = Q and ξ is a real number, the exponent (n + 1)/(4t^{2}) in the inequality (1.1) of Theorem A cannot be replaced by a real number greater than 2n/(t(t − 1)). Proposition 10.1. Let n and t be integers with 2 ≤ t ≤ n and let ξ be a complex number. There exists a constant c = c(n, t) > 0 such that, for any algebraic number α of degree n and any choice of t distinct conjugates α_{1}, …, α_{t} of α in C, we have

max_{1≤i≤t} |ξ − α_{i}| ≥ cH(α)^{−2(n−1)/(t(t−1))}.

Proof. Let P ∈ Z[T] be an irreducible polynomial of degree n, let a_{0} be its leading coefficient, and let α_{1}, …, α_{n} be the roots of P ordered so that

|ξ − α_{1}| ≤ ⋯ ≤ |ξ − α_{n}|.

Approximation by conjugate integers

Then the absolute value of the discriminant of P is the non-zero integer given by

|Disc(P)| = |a_{0}|^{2(n−1)} ∏_{1≤i<j≤n} |α_{i} − α_{j}|^{2}.

Using the estimates

|α_{i} − α_{j}| ≤ \{ \begin{matrix}2|ξ − α_{j}| ≤ 2 max\{1, |α_{i}|\} max\{1, |α_{j}|\}|ξ − α_{j}| & when 1 ≤ i < j ≤ t, \\ |α_{i}| + |α_{j}| ≤ 2 max\{1, |α_{i}|\} max\{1, |α_{j}|\} & otherwise, \end{matrix}

we find

|Disc(P)| ≤ 2^{n(n−1)}M(P)^{2(n−1)} ∏_{j=1}^{t} |ξ − α_{j}|^{2(j−1)},

where M(P) = |a_{0}| ∏_{i=1}^{m} max{1, |α_{i}|} denotes the Mahler measure of P. Since M(P) ≤ (n + 1)^{1/2}H(P) (see Theorem 2.8 in Chapter 3 of [9]), this gives

1 ≤ 2^{n(n−1)}(n + 1)^{n−1}H(P)^{2(n−1)}|ξ − α_{t}|^{t(t−1)}

and the conclusion follows.

Our last result justifies the remark made in the introduction concerning simultaneous approximation of several numbers by conjugate algebraic numbers.

Proposition 10.2. Let n and t be positive integers, and let κ be a real number with

κ > t^{−1}(t + 1)^{1+(1/t)}.

Then, there exist real numbers ξ_{1}, …, ξ_{t} and a constant H_{0} ≥ 1 (depending on n, t, κ) with the following property. For any real number H ≥ H_{0} and any choice of complex algebraic numbers α_{1}, …, α_{t} ∈ C of degree ≤ n and height ≤ H, we have

max_{1≤j≤t} |ξ_{j} − α_{j}| ≥ H^{−κn^{1/t}}.

Note that t^{−1}(t + 1)^{1+(1/t)} is a decreasing function of t for t > 0 tending to 1 as t tends to infinity. For t ≥ 2, one can take κ = 3.

Proof. Put b = (t + 1)n and define a sequence of positive integers (a_{ℓ})_{ℓ≥1} by the formula a_{ℓ} = [b^{ℓ/t}]. We claim that the real numbers

ξ_{j} = ∑_{i=0}^{∞} 2^{−a_{j+ti}}, (j = 1, …, t),

Approximation by conjugate integers

have the required property.

To prove this, we choose a real number ϵ with 0 < ϵ < 1 such that

κ > \frac{t + 1 + ϵ}{t − ϵ}(t + 1)^{1/t},

and consider the sequence of closed intervals (I_{ℓ})_{ℓ≥1} of R given by

I_{ℓ} = [κ^{−1}(t + 1 + ϵ)a_{ℓ}n^{1−(1/t)}, (t − ϵ)a_{ℓ}n] .

Two consecutive such intervals I_{ℓ} and I_{ℓ+1} overlap for sufficiently large values of ℓ since

lim_{ℓ→∞} \frac{(t − ϵ)a_{ℓ}n}{κ^{−1}(t + 1 + ϵ)a_{ℓ+1}n^{1−(1/t)}} = \frac{(t − ϵ)κ}{(t + 1 + ϵ)(t + 1)^{1/t}} > 1.

Therefore, the union of these intervals contains a half line [c, ∞) for some constant c > 0. Choose a real number H with log_{2} H ≥ c where log_{2} H denotes the logarithm of H in base 2, and let α_{1}, …, α_{t} ∈ C be algebraic numbers of degree ≤ n and height ≤ H. By definition of c, there exists an integer ℓ ≥ 1 such that log_{2} H ∈ I_{ℓ}. Writing ℓ in the form ℓ = j + tm for some integers j and m with 1 ≤ j ≤ t and m ≥ 0, we claim more precisely that, if H is sufficiently large (so that ℓ is large), we have

|ξ_{j} − α_{j}| ≥ H^{−κn^{1/t}}.

To prove this refined claim, we consider the rational number

r = ∑_{i=0}^{m} 2^{−a_{j+ti}}.

Since (a_{j+ti})_{i≥0} is a strictly increasing sequence of positive integers, it satisfies

H(r) = 2^{a_{ℓ}} and 2^{−a_{ℓ+t}} ≤ ξ_{j} − r = ∑_{i=1}^{∞} 2^{−a_{ℓ+ti}} ≤ 2^{−a_{ℓ+t}+1}.

If α_{j} = r, we find

|ξ_{j} − α_{j}| = ξ_{j} − r ≥ 2^{−a_{ℓ+t}} ≥ H^{−κn^{1/t}}

assuming that H (and thus ℓ) is sufficiently large so that

a_{ℓ+t} ≤ (t + 1 + ϵ)na_{ℓ} ≤ κn^{1/t} log_{2} H.

Approximation by conjugate integers

If α_{j} ≠ r, Liouville’s inequality gives

|α_{j} − r| ≥ γH(α_{j})^{−1}H(r)^{−n} ≥ γH^{−1}2^{−a_{ℓ}n},

with γ = γ(n) = 2^{1−n}(n + 1)^{−1/2}. This implies

|α_{j} − r| ≥ γ2^{−(t−ϵ)a_{ℓ}n−a_{ℓ}n} ≥ 2^{−a_{ℓ+t}+2} ≥ 2|ξ_{j} − r|,

assuming that H is sufficiently large so that

(t + 1 − ϵ)a_{ℓ}n ≤ a_{ℓ+t} − 2 + log_{2}(γ).

Since κ > (t + 1 + ϵ)/(t + ϵ), we may also assume (γ/2)H^{−1} ≥ H^{−((t+ϵ)/(t+1+ϵ))κn^{1/t}} and so we get

|ξ_{j} − α_{j}| ≥ |α_{j} − r| − |ξ_{j} − r| ≥ \frac{1}{2}|α_{j} − r| ≥ \frac{γ}{2} H^{−1}2^{−a_{ℓ}n} ≥ H^{−κn^{1/t}}.

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[3] Bombieri, E.; Vaaler, J. – On Siegel’s lemma. Invent. Math. 73 (1983), 11–32.

[4] Davenport, H.; Schmidt, W.M. – Approximation to real numbers by algebraic integers. Acta Arith. 15 (1969), 393-416.

[5] Cartan, H. – Cours de calcul différentiel. Hermann, Paris, 1977.

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[7] Cassels, J. W. S. – Global fields. Chapter II in: Algebraic number theory, J. W. S. Cassels and A. Fröhlich editors, Academic Press, 1967.

[8] Fröhlich, A.; Taylor, M. J. — Algebraic number theory. Cambridge Studies in Advanced Mathematics, 27. Cambridge University Press, Cambridge, 1993.

Approximation by conjugate integers

[9] Lang, S. – Fundamentals of Diophantine geometry. Springer-Verlag, New-York, 1983.

[10] Laurent, M.; Roy, D. – Criteria of algebraic independence with multiplicities and interpolation determinants. Trans. Amer. Math. Soc. 351 (1999), 1845-1870.

[11] Mahler, K. – Ein Übertragungsprinzip für konvexe Körper. Časopis Pěst. Mat. 68 (1939), 93–102.

[12] Mahler, K. – Inequalities for ideal bases in algebraic number fields, J. Austral. Math. Soc. 4 (1964), 425–448.

[13] Macfeat, R. B. – Geometry of numbers in adele spaces. Dissertationes Math. (Rozprawy Mat.) 88 (1971), 54 pp.

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Damien ROY Département de Mathématiques Université d’Ottawa 585 King Edward Ottawa, Ontario K1N 6N5, Canada droy@uottawa.ca http://aix1.uottawa.ca/∼droy/ Michel WALDSCHMIDT Université P. et M. Curie (Paris VI) Institut de Mathématiques CNRS UMR 7586 Théorie des Nombres, Case 247 175, rue du Chevaleret, 75013 Paris, France miw@math.jussieu.fr http://www.math.jussieu.fr/∼miw/