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The Skolem-Abouzaid theorem in the singular case

Let F(X;Y) in Q[X;Y] be a Q-irreducible polynomial. In 1929 Skolem proved the following theorem:"Assume that F(0;0) = 0. Then for every non-zero integer d, the equation F(X;Y) = 0 has only finitely many solutions in integers (X;Y) with gcd(X;Y) = d". Skolem method allows one to bound the solutions explicitly in terms of the coefficients of the polynomial F and the integer d. In 2008, Abouzaid gave a far-going generalization of Skolem theorem. He extended it in two directions: first, he studied solutions not only in rational integers, but in arbitrary algebraic numbers. Second, he not only bounded the solution in terms of the logarithmic gcd, but obtained a sort of asymptotic relation between the heights of the coordinates and their logarithmic gcd. Unfortunately, Abouzaid assumption is slightly more restrictive than Skolem: he assumes not only that the point (0;0) belongs to the plane curve F(X;Y) = 0, but that (0;0) is a non-singular point on this curve. The purpose of the present article is to get rid of this non singularity hypothesis.

In the same year, Siegel obtained his celebrated finiteness theorem for integral solutions of Diophantine equations: equation F (X, Y ) = 0 has finitely many solutions in integers unless the corresponding plane curve is of genus 0 and has at most 2 points at infinity. While Siegel's result is, certainly, deeper and more powerful than Theorem 1.1, the latter has one important advantage. Siegel's theorem is known to be non-effective: it does not give any bound for the size of integral solutions. On the contrary, Skolem's method allows one to bound the solutions explicitly in terms of the coefficients of the polynomial F and the integer d. Indeed, such a bound was obtained by Walsh [9]; see also [4].
In 2008 Abouzaid [1] gave a far-going generalization of Skolem's theorem. He extended it in two directions.
First, he studied solutions not only in rational integers, but in arbitrary algebraic numbers. To accomplish this, he introduced the notion of logarithmic gcd of two algebraic numbers α and β which coincides with the logarithm of the usual gcd when α, β ∈ Z.
Second, he not only bounded the solution in terms of the logarithmic gcd, but obtained a sort of asymptotic relation between the heights of the coordinates and their logarithmic gcd.
Let us state Abouzaid's principal result (see [1,Theorem 1.3]). In the sequel we assume that F (X, Y ) ∈Q[X, Y ] is an absolutely irreducible polynomial, and use the notation m = deg X F, n = deg Y F, M = max{m, n}. ( We denote by h(α) the absolute logarithmic height of α ∈Q and by lgcd(α, β) the logarithmic gcd of α, β ∈ Q. We also denote by h p (F ) the projective height of the polynomial F . For all definitions, see Subsection 3.1.
Informally speaking, as max{h(α), h(β)} → ∞. Unfortunately, Abouzaid's assumption is slightly more restrictive than Skolem's (1): he assumes not only that the point (0, 0) belongs to the plane curve F (X, Y ) = 0, but that (0, 0) is a non-singular point on this curve. The purpose of the present article is to get rid of this nonsingularity hypothesis.
Let us define, first of all, a certain quantity r which would "measure the singularity" of the point (0, 0). Let C be (a non-singular projective model of) the plane algebraic curve defined by F (X, Y ) = 0. We denote by x, y the coordinate functions 1 on the curve C. We set where the sum runs over the points of C with x(P ) = y(P ) = 0. Clearly, r > 0 if and only if F (0, 0) = 0 and r = 1 if and only (0, 0) is a non-singular point of the plane curve F (X, Y ) = 0.
We can now state our principal result.

Definitions and results used in the article
In this section, we compile some definitions and results from different sources, which will be required for our proof. We advise the reader interested in the proof to go directly to Section 3 and come back to this Section 2 as required.
We normalize the absolute values on number fields so that they extend standard absolute values on Q: if v | p (non-Archimedean) then |p| v = p −1 and if v | ∞ (Archimedean) then |2014| v = 2014. We denote by M K the set of places (normalized absolute values) of the number field K.

Heights and lgcd of algebraic numbers
The height of an algebraic number α is defined as where K is a number field containing α and log + = max{log, 0}. It is well-known that the height does not depend on the particular choice of K, but only on the number α itself. It is equally well-known that h(α) = h(α −1 ), so that where log − = min{log, 0} and The quantities h v (α) can be viewed as "local heights". Clearly, h v (α) ≥ 0 for any v and α.
We define the logarithmic gcd of two algebraic numbers α and β, not both 0, as where K is a number field containing both α and β. It again depends only α and β, not on K. A simple verification shows that for α, β ∈ Z we have lgcd(α, β) = log gcd(α, β). Now let K be a number field and S be a set of places of K. We define the S-height by Similarly we define lgcd S . We shall frequently use the inequality lgcd S (α, β) ≤ h S (α) ≤ h(α) without special reference.

Affine and projective heights of polynomials
We define the projective and the affine height of a vector a = (a 1 , · · · , a m ) ∈Q with algebraic entries by Here, K is a number field containing a 1 , · · · , a m , and d, d v are defined as in the previous subsection. We notice that the height of an algebraic number defined in the previous subsection corresponds to the affine height of a projective vector.
We define the projective and affine height of a polynomial as the corresponding heights of the vector of its non-zero coefficients. If F is a non-zero polynomial, then: and h p (F ) = h a (F ) if F has a coefficient equal to 1. In [6, Lemma 4], Schmidt proves the following lemma: be a polynomial with algebraic coefficients, such that m = deg X F and n = deg Y F . Let R F (X) = Res Y (F, F Y ) be the resultant of F and its derivative polynomial with respect to Y. Then: Another useful theorem on the height of the resultants is [1, Proposition 2.4]: Proposition 2.2 Let F 1 (X, Y ) and F 2 (X, Y ) be polynomials with algebraic coefficients, and let R(X) be their resultant with respect to the variable Y . Put: Then h p (R) ≤ n 1 h p (F 2 ) + n 2 h p (F 1 ) + (m 1 n 2 + m 2 n 1 ) + (n 1 + n 2 ) log(n 1 + n 2 ).

Coefficients versus roots
In this subsection we establish some simple relations between coefficients and roots of a polynomial over a field with absolute value, needed in the proof of our main result. It will be convenient to use the notion of v-Mahler measure of a polynomial. Let K be a field with absolute value v and f (X) ∈ K[X] a polynomial of degree n. Let β 1 , . . . , β n ∈K be the roots of f : f (X) = a n X n + a n−1 X n−1 + · · · + a 0 = a n (X − β 1 ) · · · (X − β n ).
Define the v-Mahler measure of f by where we extend v somehow toK. (Clearly, M v (f ) does not depend on the particular extension of v.) It is well-known that |f | v = M v (f ) for non-archimedean v ("Gauss lemma") and Lemma 2.5 Let β 1 , . . . , β +1 be + 1 distinct roots of f (X), where 0 ≤ ≤ n − 1. Then where β 1 , . . . , β n are all roots of f (X) inK counted with multiplicities. Observe that each term in the sum above contains one of the roots β 1 , . . . , β +1 , and the product of the other roots together with a n is v-bounded by in the non-archimedean case and |a | v ≤ n µM v (f ) in the archimedean case. Since n ≤ 2 n , the result follows.
Proof Write the roots of f as β 1 , . . . , β n , where The second statement follows from Lemma 2.5: applying it with i instead of , we find as wanted.
For the first statement, using (10), we find as wanted.
We will apply Lemma 2.6 in the following situation. Let K be a field, and let F ( Viewing F (x, Y ) as a polynomial in Y over the field K((x)) of formal power series, it has n roots in the algebraic closure K((x)). We extend the additive valuation ν x from K((x)) to K((x)).

Corollary 2.7
In the above set-up, assume that among n roots y(x) of F (x, Y ) there are exactly such that ν x (y(x)) > 0. Then the polynomial F (0, Y ) has at 0 a zero of order .
This is exactly what is wanted.

Eisenstein's theorem
In this subsection, we recall the quantitative Eisentein's theorem due to work from Dwork, Robba, Schmidt and Van der Poorten, as given in [2]. It will be convenient to use the notion of M K -divisor.
An M K -divisor is an infinite vector (A v ) v∈M K of positive real numbers, each A v being associated to one v ∈ M K such that for all but finitely many v ∈ M K we have We define the height of an In [2, Theorem 7.5], Bilu and Borichev prove the following version of Eisenstein's theorem: be a power series satisfying F (X, y(X)) = 0. Then, there exists an effective for any v ∈ M K anyhow extended toK and any k ≥ κ, and such that: Applying this theorem to our case (a k = 0 for k ≤ 0) and setting κ = 0, we obtain that: for any v ∈ M K anyhow extended toK and any k ≥ 1, and such that: Another lemma that will be useful in this paper is [1, Proposition 2.7]: Lemma 2.10 Let K be a number field and let y(X) = ∞ k=1 a k X k/e be a series with coefficients in K such that there exists an effective Then, for any valuation v and for all index k ≥ 1 we have:

Siegel's "absolute" lemma
The height of a vector space W generated by the free set (w 1 , · · · , w d ) is defined as: where is a norm. In [3, Lemma 4.7], David and Philippon give the following version of Siegel's lemma: Lemma 2.11 Let W ⊂Q n+1 be a vector subspace of dimension d and let (w 1 , · · · , w d ) be a basis of W over a number field K. Then, for any ε > 0, there exists y ∈ W such that: For our purposes, we find the following corollary: Lemma 2.12 Let n be a positive integer, and Proof: Let L * = L(Q n , K) be the K-vector space of linear forms fromQ n into the base number field K. Let (L 1 , · · · , L M ) be a free system in L * . Let W = X ∈Q n : L 1 (X) = 0, · · · , L M (X) = 0 . It is well known that W is a subspace ofQ n (as the intersection of M hyperplanes). Then, the orthogonal of W , defined by W ⊥ = {Linear forms L such that ∀w i ∈ W, L(w i ) = 0}, is exactly the K-vector space generated by (L 1 , · · · , L M ). We know that, dim Clearly, Applying Lemma 2.11, we obtain that there exists y ∈ W such that: Let's set ε = 1 2d . Let L be the M × M matrix of (L 1 , · · · , L M ) expressed in (e 1 , · · · , e M ), an orthonormal basis of W ⊥ .
Hadamard's inequality applied to L gives | det L | ≤ ||L 1 || 2 · · · ||L M || 2 , which, in our case, translates into: Therefore, By taking logarithms of both sides and adding up, we obtain: We notice that if instead of a basis of W ⊥ we were working with a system (L 1 , · · · , L M )

Proof
In the cases m = 1 or n = 1 the things are much simpler, and one can easily obtain a numerically sharper result. Therefore we allow ourselves to assume that m, n ≥ 2.
Let P 1 , . . . , P s be the common zeros of x and y, and e i = ν Pi (x), κ i = ν Pi (y). Then, y has e i Puiseaux expansions in each P i belonging to the gcd of the common zeros of x and y, of the form: where κ i ≥ 1 and a (ij) κi > 0. We have κ i ≥ 1 because for all i = 1, · · · , s, we have y(P i ) = 0. We define E = s i=1 e i and Ξ = s i=1 κ i . Let K be a number field containing the coefficients of F , the numbers α, β and the coefficients a According to Lemma 2.9, there exist s effective for any v ∈ M K and any k ≥ 1.
Then, H satisfies: We know that |a if v < ∞.
Since lgcd S (α, β) = 0, we have lgcd(α, β) = lgcd S (α, β) + lgcd S (α, β). For every v ∈ S, each of the series y ij (X) converges v-adically at X = α (but not necessarily to β). The sum is a root of the polynomial F be the subset of S such that the v-adic sum of y ij (x) at α is β. We say that y ij (α) v =β, where v = means that the equality is understood in the sense of v-adic convergence. We may partition S as Then: The first term will be bounded in Proposition 3.3 and the other two in Propositions 3.4 and 3.5 respectively. From Proposition 3.3, we obtain: From Proposition 3.4, we obtain: Finally, from Proposition 3.5, we obtain: Using the bounds on H , Ξ, E and s, we obtain the sought after bounds: The rest of the proof consists in bounding these three terms. For this, we start with some preliminaries by reminding some useful definitions and results, and preparing some theorems and properties to be used in our context. Then, we easily bound the first and second term. The rest of the paper leads us to the bound of the third term.

Technical lemma
This technical lemma which will be used all throughout the rest of the proofs: (2κ i + 1) + s(m + 1)n ≤ 3mH + s(m + 1)n (using Lemma 2.9 we find that e i h(a κi ) ≤ κ i h(A (i) ), and Ξ ≤ m).

Partition of S
We remind that T ij = {v ∈ S : y ij (α) v =β}, is the subset of S such that the v-adic sum of y ij (x) at α is β (cf. 16), and S = ∪ s i=1 ∪ ei j=1 T ij ∪ T .
Proof: We have

Third bound
We now need to bound lgcd S\T (α, β) − r n h(α) . Remember that S \ T = ∪ s i=1 ∪ ei j=1 T ij where T ij has been defined in (16).
From now on, we focus on a particular T ij . We fix i and j, and write T = T ij , y(X) = y ij (X), k , a κ = a κi · · · Proposition 3. 6 We have Proof: k=κ a k α k/e . According to Lemma 2.9, there exists an effective Let's call (A) the assumption " κ e ≥ 1" and (B) the assumption " κ e < 1".
In the non-Archimedean case, we know that v ∈ S ⇒ |a k α k/e | v −→ k→∞ 0 and thus that ∞ k=κ a k α k/e converges v-adically. We just proved that in this case, we also have ∀k ≥ κ, |a k α| In the Archimedean case, we have obtained that: We notice that these results remain valid in Case (B). Thus, if v < ∞, and therefore, Summing up over v ∈ T , we obtain: Case (B) We have: However, for k ≥ κ + 1: if v < ∞.
In the non-Archimedean case, we know that v ∈ S ⇒ |a k α k/e | v −→ k→∞ 0 and thus that ∞ k=κ+1 a k α k/e converges v-adically. As we just proved that in this case, we also have ∀k > κ, |a k α k/e | v < |a κ α κ/e | v , thus | ∞ k=κ+1 a k α k/e | v < |a κ α κ/e | v . Therefore, In the Archimedean case, we have obtained that: Using the triangular inequality in this case, we have that: Thus, We have hence bounded |β| v both in the Archimedean and in the non-Archimedean cases: We can now bound the quantity we are interested in, that is Summing up over v ∈ T , we obtain: We know that h(a −1 κ ) = h(a κ ) and using Lemma 2.9, we find that h(a κ ) ≤ κ e h(A ) Therefore, in case (B), we obtain: Thus, combining both cases we obtain: Proposition 3.7 Let 0 < ε ≤ 1 2 . Then, we have either: Proof We split the proof of Proposition (3.7) in two Propositions (3.9 and 3.10), proved a little further, to find first an upper bound and then a lower bound. For this, we shall need an auxiliary polynomial with very useful properties (Proposition 3.8). Combining Propositions (3.9) and (3.10), we obtain Proposition (3.7). Let us fix δ satisfying 0 < δ ≤ 1 2 and N > 1.
Condition η = ν X (G(X, y(X))) ≥ (1 − δ)N n is equivalent to a system of at most (1 − δ)N n equations in the coefficients of G. Each coefficient of these equations can be viewed as a linear form with algebraic coefficients a ( ) k (like in Lemma 2.10, 0 ≤ ≤ n − 1, k ≤ (1 − δ)N n), the variables being the coefficients of G. We write, G(X, y(X)) = ∞ k=0 G k X k/e = ∞ k=η G k X k/e . Lemma 2.10 implies that the height of each linear equation is bounded by nN (h(A ) + log 4) (we use the trivial estimates l+k−1 k ≤ 2 +k−1 ≤ 4 nN ). Using Lemma 2.12, there exists G with deg G = N (n − 1) such that: Applying Lemmata 2.10 and 2.9, we find that: We may assume that at least one coefficient of G is one, thus the affine height is the same as projective height. For the sake of notation simplification, let γ = G(α, β).
Let us obtain a lower bound for h(γ). For that, let us first define a new set: In this case, ≤ 4nh p (F ) + 3n log(nm) + 13ne.
We know that G(X, y(X)) = ∞ k=η G k X k (where η ≥ (1 − δ)N n). Then, for all v ∈ T , we have G(α, y(α)) v =γ. We observe that each G k is a sum of at most nN terms in g k and a k . Applying Lemma 2.10 and using the bound n(N + 1) +k−1 k ≤ 6 k because k ≥ η ≥ 1 2 nN , we find: Therefore, and thus, taking log + of both sides and summing over v ∈ T * , h T * (γ) ≥ ηh T * (α) − ηh(A ) − h p (G) − η log 6.