aa r X i v : . [ m a t h . N T ] S e p BIVARIATE POLYNOMIAL INJECTIONS AND ELLIPTIC CURVES
HECTOR PASTEN
Abstract.
For every number field k , we construct an affine algebraic surface X over k with aZariski dense set of k -rational points, and a regular function f on X inducing an injective map X ( k ) → k on k -rational points. In fact, given any elliptic curve E of positive rank over k , we cantake X = V × V with V a suitable affine open set of E . The method of proof combines valuedistribution theory for complex holomorphic maps with results of Faltings on rational points insub-varieties of abelian varieties. Introduction
Results.
It is not known whether there is a bivariate polynomial f ∈ Q [ x , x ] inducing aninjective function Q × Q → Q . According to Remarque
10 in [6] (which dates back to 1999)the existence of such an f was first asked by Harvey Friedman, and Don Zagier suggested that f = x + 3 x should have this property. In the direction of a positive answer to Friedman’squestion, we prove an injectivity result for bivariate polynomial functions on elliptic curves; in fact,our polynomial functions are very close to Zagier’s suggestion. Theorem 1.1.
Let k be a number field. Let a, b ∈ k with a + 27 b = 0 and consider the smoothaffine curve C ⊆ A k defined over k by the equation y = x + ax + b. Let α, β, γ ∈ k with α, β = 0 and γ / ∈ {− , , } . Let n ≥ be a positive integer such that the only n -th root of unity in k is . Define the morphism P : C → A k by P ( x, y ) = αx + βy and define themorphism f : C × C → A k by f (( x , y ) , ( x , y )) = P ( x , y ) n + γ · P ( x , y ) n = ( αx + βy ) n + γ ( αx + βy ) n . We have the following: (i)
The map C ( k ) → k induced by P on k -rational points is injective away from finitely manypoints of C ( k ) . (ii) Let U ⊆ C be any non-empty Zariski open set defined over k such that the function U ( k ) → k induced by P is injective. Then the function U ( k ) × U ( k ) → k induced by f is injectiveaway from finitely many k -rational points of U ( k ) × U ( k ) . The previous theorem will be obtained in Section 4 an application of Theorem 3.2 together withTheorem 2.1 and Proposition 3.1.A striking consequence of Theorem 1.1 is the following phenomenon:
For every number field k there is an affine algebraic variety X of dimension greater than (in fact, a surface) with amorphism f : X → A k defined over k such that the k -rational points of X are dense in X , andnonetheless f induces an injective function X ( k ) → k on k -rational points. More precisely, we havethe following consequences of Theorem 1.1. We refer to Section 4 for details.
Date : September 5, 2019.2010
Mathematics Subject Classification.
Primary 11G05; Secondary 11G35, 30D35.
Key words and phrases.
Injective, bivariate polynomial, elliptic curve, uniqueness polynomial.This research was supported by FONDECYT Regular grant 1190442. orollary 1.2. Let k be a number field and let E be an elliptic curve over k of positive Mordell-Weilrank. There is a non-empty affine open set V ⊆ E defined over k and a morphism f : V × V → A k also defined over k such that the k -rational points of the affine surface V × V are dense, and f induces an injective map V ( k ) × V ( k ) → k on k -rational points. Corollary 1.3.
Let C ⊆ A Q be the affine curve defined over Q by y = x + x − . Let f be thepolynomial function on C × C defined by f (( x , y ) , ( x , y )) = ( x + y ) + 2( x + y ) . In the real topology, the Q -rational points of the algebraic surface C × C are dense in C ( R ) × C ( R ) ,and the latter is homeomorphic to R .Furthermore, the function C ( Q ) × C ( Q ) → Q induced by f is injective away from finitely many Q -rational points of C × C . Corollary 1.3 shows, in particular, that there is no topological obstruction to a possible positiveanswer to Friedman’s question.An important tool in our approach is the concept of strong uniqueness function for holomorphicmaps from C to complex elliptic curves, which we introduce and study in Section 2. Then we applythem in Section 3 to explicitly compute the Zariski closure of rational points in certain varietiesassociated to injectivity problems.Since we work with elliptic curves, the relevant varieties in our arguments appear as sub-varietiesof abelian varieties. Results of Faltings allow us to study the rational points of these varieties provided that we can explicitly compute the (translates of) positive dimensional abelian varietiescontained in them. It is at this point where strong uniqueness functions are used, together withresults of Hayman on holomorphic maps to Fermat varieties.1.2. Previous work.
As far as we know, our results give the first unconditional progress onFriedman’s question over Q and number fields. Besides this, there are two other results conditionalon standard conjectures in Diophantine Geometry:In 1999, Cornelissen [6] observed that the 4-terms abc -conjecture [4] implies that polynomialsof the form x n + 3 y n induce injections Q × Q → Q if n is odd and large enough. The basic ideais that failure of injectivity gives rise to a non-degenerate solution of the 4-term Fermat equation a n + 3 b n = c n + 3 d n where the 4-terms abc -conjecture can be applied. Cornelissen also proved anunconditional analogue over function fields, where the 4-terms abc -conjecture can be replaced by atheorem of Mason.In 2010, Poonen [16] proved that the Bombieri-Lang conjecture on rational points of surfaces ofgeneral type also implies a positive answer to Friedman’s question.Let us briefly recall the main ideas in Poonen’s work [16]. He observed that when f ( x, y ) ∈ Q [ x, y ]is homogeneous, it induces an injective map Q × Q → Q if and only if the Q -rational points of theprojective surface Z = { f ( x , x ) = f ( x , x ) } ⊆ P Q lie in the line L = { x = x , x = x } ⊆ Z . Forgeneral f with deg( f ) ≥
5, the Bombieri-Lang conjecture for surfaces implies that the Q -rationalpoints of the surface Z are algebraically degenerate, but this is not enough for injectivity as theZariski closure of the rational points in Z might contain other components besides L . While it isunclear how to exactly compute the Zariski closure of the rational points even under the Bombieri-Lang conjecture, Poonen managed to produce a suitably ramified cover of Z to get rid of thoseother components (if any), leading to a new polynomial f ( x, y ) ∈ Q [ x, y ] of larger degree whichhas the desired injectivity property conditional on the Bombieri-Lang conjecture.Our approach of course owes some ideas to the work of Cornelissen (e.g. the use of 4-terms Fermatequations as part of the construction) and Poonen (e.g. to formulate the injectivity problem as aproblem about rational points in varieties). However, there are a number of crucial differences. Forinstance, in our context we need to consider higher dimensional varieties —not just surfaces— and e need to explicitly determine the Zariski closure of the relevant rational points. Our method forcomputing this Zariski closure pertains to the theory of value distribution of complex holomorphicmaps.1.3. Additional motivation.
Besides arithmetic interest, there are other motivations in the lit-erature for studying bivariate polynomial injections.In 1895, Cantor [5] showed that the polynomial f ( x, y ) = 12 ( x + y )( x + y + 1) + y defines a bijection Z ≥ × Z ≥ → Z ≥ and used this fact to prove that Z ≥ is countable. Thepolynomial f ( x, y ) = f ( y, x ) has the same property, and its is an old open problem whether f and f are the only two polynomials inducing a bijection Z ≥ × Z ≥ → Z ≥ . In 1923, Fueterand Polya [10] proved that this is the case among quadratic polynomials. In 1978, the result wasextended to polynomials of degree at most 4 by Lew and Rosenberg [14, 15].Bivariate injections Z × Z → Z are known as storing functions in computability theory. Cornelis-sen [6] extended this notion to a model-theoretic context by introducing the concept of positiveexistential storing; his study of Friedman’s question was developed in this setting.More recently, the topic of bivariate polynomial injections has received attention for crypto-graphic purposes. Boneh and Corrigan-Gibbs [3] developed a new commitment scheme (amongother applications) motivated by the conjecture that some bivariate polynomial f ∈ Q [ x , x ] suchas Zagier’s polynomial x + 3 x defines an injective map Q → Q . Attacks on this commitmentscheme have been studied by Zhang and Wang [17].2. Strong uniqueness functions on elliptic curves
Let M be the field of (possibly transcendental) complex meromorphic functions on C . We recallthat a polynomial P ( x ) ∈ C [ x ] is said to be a uniqueness polynomial if the equation P ( f ) = P ( f )with f , f ∈ M non-constant implies f = f . On the other hand, a polynomial P ( x ) ∈ C [ x ] is saidto be a strong uniqueness polynomial if the equation P ( f ) = cP ( f ) with f , f ∈ M non-constantand c ∈ C × implies f = f (in particular, it implies c = 1). Uniqueness polynomials and stronguniqueness polynomials are a classic topic in value distribution theory of holomorphic maps, andthere is abundant literature on this subject, see for instance [1, 2, 11, 13].Non-constant elements of M are the same as holomorphic maps from C to P , and by Picard’stheorem the only other algebraic curves admitting non-constant holomorphic maps from C areelliptic curves. Let us introduce a notion of (strong) uniqueness function for the latter setting; suchfunctions will naturally arise in our study of arithmetic injectivity problems.Let E be a complex elliptic curve. A rational function P ∈ K ( E ) is a uniqueness function on E if the equation P ( f ) = P ( f ) with f , f : C → E non-constant holomorphic maps implies f = f . On the other hand, a rational function P ∈ K ( E ) is a strong uniqueness function on E if the equation P ( f ) = cP ( f ) with f , f : C → E non-constant holomorphic maps and c ∈ C × implies f = f (in particular, it implies c = 1). It turns out that every complex elliptic curveadmits strong uniqueness functions of a very simple kind. Theorem 2.1.
Let E be an elliptic curve over C , choose a Weierstrass equation y = x + Ax + B for E and define the rational functions x, y ∈ K ( E ) as corresponding coordinate projections. Let α, β ∈ C be non-zero, and consider the non-constant rational function P = αx + βy ∈ K ( E ) . Then P is a strong uniqueness function on E .Remark . The conditions α = 0 and β = 0 are necessary: For any complex elliptic curve E andany non-constant holomorphic map f : C → E we have x ( f ) = x ([ − ◦ f ) and y ( f ) = − · y ([ − ◦ f ) . emark . The conditions α = 0 and β = 0 are necessary even if we work with uniquenessfunctions. This is clear for β = 0. For α = 0 we can choose the elliptic curve E with affine equation y = x + 1. We consider the automorphism u of E defined by ( x, y ) ( ǫx, y ) with ǫ a primitivecubic root of 1. Let us take any non-constant holomorphic map f : C → E and define f = u ◦ f .Then y ( f ) = y ( u ◦ f ). Remark . If Q is a strong uniqueness polynomial for M and P is a uniqueness function on anelliptic curve E , then Q ◦ P is a strong uniqueness function on E . Thus, Theorem 2.1 togetherwith standard results in the theory of strong uniqueness functions, allow one to construct stronguniqueness functions on E of arbitrarily large degree. Remark . After learning about this work, Michael Zieve managed to prove (private communica-tion) that if E is given in Weierstrass form as in Theorem 2.1 with coordinates x and y , then “most”polynomials P ( x, y ) ∈ C [ x, y ] give strong uniqueness functions on E . Thus, strong uniqueness func-tions are abundant. The linear case given in Theorem 2.1 suffices for our purposes, although a moregeneral description of strong uniqueness functions might be useful for other diophantine problems.In preparation for the proof of Theorem 2.1, we record here the following simple lemma. Lemma 2.2.
Let E be an elliptic curve over C . (a) Let θ : C → E be a non-constant holomorphic map. Then θ is surjective. (b) Let ψ : C → E × E be a non-constant holomorphic map with algebraically degenerate image.Then the image of ψ is exactly the translate of an elliptic curve subgroup of E × E .Proof. For (a), let us choose a uniformization q : C → E and a holomorphic lift ˜ θ : C → C satisfying q ◦ ˜ θ = θ . The entire function ˜ θ has at most one exceptional value in C by Picard’s theorem, hence θ is surjective.Item (b) is a special case of the literature around Bloch’s conjecture, but this case admits adirect proof: The Zariski closure of the image of ψ is an algebraic curve Y in E × E of geometricgenus g = 1 ( g = 0 is excluded by the Riemann-Hurwitz theorem, and g ≥ E × E → E we conclude that Y is the translate of anelliptic curve subgroup of E × E . By part (a), ψ is surjective onto Y . (cid:3) For a complex elliptic curve E we denote its neutral element by 0 E . Proof of Theorem 2.1.
Let c ∈ C × and let f , f : C → E be non-constant holomorphic mapssatisfying P ◦ f = c · P ◦ f . Observe that both P ◦ f , P ◦ f ∈ M are non-constant. We will provethat f = f , which implies c = 1.Let us define the holomorphic map φ : C → E × E, φ ( z ) = ( f ( z ) , f ( z )) . We observe that φ does not have Zariski dense image in E × E . This is because P : E → P inducesa finite morphism P × P : E × E → P × P , and the condition P ◦ f = c · P ◦ f implies that theimage of φ is contained in the pull-back of { ([ s : ct ] , [ s : t ]) : [ s : t ] ∈ P } ⊆ P × P under P × P .Therefore φ has algebraically degenerate image in E × E . By part (b) of Lemma 2.2, φ surjectsonto an algebraic curve Y ⊆ E × E which is the translate of an elliptic curve subgroup in E × E .We claim that that (0 E , E ) ∈ Y , i.e. that Y is in fact an elliptic curve subgroup of E × E . Thisis because the only pole of P = αx + βy occurs at 0 E , which is in the image of both f and f bypart (a) of Lemma 2.2. Since P ◦ f = c · P ◦ f with c ∈ C × , it follows that f − (0 E ) = f − ( P − ( ∞ )) = ( P ◦ f ) − ( ∞ ) = ( P ◦ f ) − ( ∞ ) = f − ( P − ( ∞ )) = f − (0 E )which is non-empty. This proves (0 E , E ) ∈ Y . herefore, the image of φ = ( f , f ) : C → E × E is an elliptic curve subgroup Y ⊆ E × E , andby considering the projections p , p : E × E → E we see that Y is isogenous to E .Fix an isogeny v : E → Y and let u j = p j ◦ v ∈ End( E ) for j = 1 ,
2. Since v is ´etale, there is aholomorphic map h : C → E that lifts φ via v in the sense that φ = v ◦ h . Since φ = ( f , f ), weobserve that f j = p j ◦ φ = p j ◦ v ◦ h = u j ◦ h for j = 1 ,
2. It suffices to show u = u .For a suitable lattice Λ ⊆ C (depending on our choice of Weierstrass equation for E ), theassociated Weierstrass function ℘ ∈ M induces an explicit uniformization of w : C → E , as itsatisfies (cid:18) ℘ ′ ( z ) (cid:19) = ℘ ( z ) + A℘ ( z ) + B. The unformization w : C → E is determined by ℘ in the sense that(2.1) x ◦ w = ℘ and y ◦ w = 12 ℘ ′ . There are non-zero complex numbers λ , λ ∈ C × lifting the endomorphisms u and u via w , sothat ( u j ◦ w )( z ) = w ( λ j · z ). It suffices to prove λ = λ , as this will show u = u , hence, f = f .Also, we recall that the Laurent expansions of ℘ ( z ) and ℘ ′ ( z ) near z = 0 are ℘ ( z ) = 1 z + X j ≥ c j z j and ℘ ′ ( z ) = − z + X j ≥ jc j z j − for suitable complex numbers c j ∈ C . Let us choose a lift ˜ h : C → C of the holomorphic map h : C → E via the uniformization w , so that h = w ◦ ˜ h . Furthermore, since h : C → E is surjective(cf. item (a) in Lemma 2.2) we can choose the holomorphic lift ˜ h : C → C in such a way that 0 isin its image.Recalling that P = αx + βy and that f j = u j ◦ h , the condition P ◦ f = c · P ◦ f becomes α · x ◦ u ◦ h + β · y ◦ u ◦ h = cα · x ◦ u ◦ h + cβ · y ◦ u ◦ h. From the relation ( u j ◦ h )( z ) = ( u j ◦ w ◦ ˜ h )( z ) = w ( λ j · ˜ h ( z )) together with (2.1), we deduce α · ℘ ( λ · ˜ h ( z )) + β · ℘ ′ ( λ · ˜ h ( z )) = cα · ℘ ( λ · ˜ h ( z )) + cβ · ℘ ′ ( λ · ˜ h ( z )) . Since 0 is in the image of ˜ h : C → C , we have that the image of ˜ h contains a neighborhood of0 ∈ C . From the previous relation we deduce that the following holds for the complex variable z ina neighborhood of 0 : α · ℘ ( λ · z ) + β · ℘ ′ ( λ · z ) = cα · ℘ ( λ · z ) + cβ · ℘ ′ ( λ · z ) . Considering the coefficients of z − and z − in the Laurent expansion near z = 0, we deduce βλ − = cβλ − and αλ − = cαλ − . Recall that c, α, β, λ , λ are non-zero complex numbers. We finally deduce λ = βα · cαλ − cβλ − = βα · αλ − βλ − = λ . (cid:3) . Arithmetic results If X and Y are algebraic varieties over a field k we write X × Y for X × k Y , and the diagonalin X × X is denoted by ∆ X .The following proposition is an application of Faltings theorem for rational points on curves ofgenus at least 2 and the notion of uniqueness function. The proof can be useful to illustrate theideas in the proof of our main injectivity results for products of elliptic curves (cf. Theorem 3.2). Proposition 3.1.
Let k be a number field. Let E be an elliptic curve over k and let P ∈ k ( E ) bea uniqueness function defined over k . Let V ⊆ E be the locus where P is regular. The function V ( k ) → k induced by P is injective away from finitely many points of V ( k ) .Proof. Using the morphism P × P : V → A k , we let Z ⊆ V × V be defined by ( P × P ) ∗ ∆ A k . Let Z ⊆ E be the Zariski closure of Z and observe that Z − Z consists of finitely many points. Wenote that Z is a finite union of curves of geometric genus at least 1, since it is a divisor on E , andthe diagonal ∆ E is one of these curves.We claim that the only irreducible component of Z of geometric genus 1 is ∆ E . In fact, let Y beany irreducible component of Z with geometric genus 1, then there is a non-constant holomorphicmap h : C → E whose image is contained in Y . We write h = ( h , h ) with h j : C → E holomorphic and at least one of them non-constant. Let us observe that P ◦ h = P ◦ h because Y is a component of Z and Z − Z consists of finitely many points. It follows that both h and h are non-constant, and since P is a uniqueness function we deduce that h = h . Hence Y = ∆ E .Finally, note that we must prove that only finitely many pairs ( x, y ) ∈ ( V − ∆ E )( k ) satisfy P ( x ) = P ( y ), which is the same as proving that only finitely k -rational points of Z lie outside ∆ E .The result follows by Faltings theorem for curves [7] and our previous genus computation. (cid:3) The following is our main tool for proving the injectivity results stated in the introduction.
Theorem 3.2.
Let k be a number field. Let E be an elliptic curve over k and let P ∈ k ( E ) be astrong uniqueness function defined over k . Let U ⊆ E be a non-empty Zariski open subset of E defined over k satisfying that P is regular on U and that the function U ( k ) → k induced by P isinjective. Let n ≥ be a positive integer such that the only n -th root of unity in k is . Let γ ∈ k be different from − , , and . Let f : U × U → A k be the morphism defined over k by f ( q , q ) = P ( q ) n + γ · P ( q ) n . The function U ( k ) × U ( k ) → k induced by f is injective away from finitely many k -rational pointsof U × U .Remark . The function P required by Theorem 3.2 exists for every choice of E ; our Theorem2.1 gives an assortment of them. Furthermore, Proposition 3.1 shows that the open set U requiredby Theorem 3.2 always exists. Nevertheless, in concrete cases one might explicitly construct suchan open set U by other means such as a local argument, cf. the proof of Corollary 1.3 in Section 4for instance. Proof of Theorem 3.2.
Let Z ⊆ U be the pull-back of ∆ A k by the morphism f × f : U → A k .Since f is a finite map, Z is a divisor in U defined over k . Also, we have ∆ U ⊆ Z .It suffices to show that all but finitely many k -rational points of Z are contained in ∆ U .Let Z be the Zariski closure of Z in E . Let D = Z − Z . Note that the Zariski closure of∆ U ⊆ U = U × U in E is precisely ∆ E . It suffices to show that Z − ( D ∪ ∆ E ) contains atmost finitely many k -rational points.By Faltings theorem on sub-varieties of abelian varieties [8, 9], the (finitely many) irreduciblecomponents of the Zariski closure of Z ( k ) are translates of abelian sub-varieties of E defined over . Let Y be any of these irreducible components with strictly positive dimension. It suffices toshow that Y ⊆ D ∪ ∆ E , and we will prove this by contradiction.For the sake of contradiction, let us suppose that Y is not contained in D ∪ ∆ E . Since Y is thetranslate of a positive-dimensional abelian sub-variety of E , there is a non-constant holomorphicmap h : C → Y with Zariski dense image (this can be seen by realizing Y as a quotient of C d bysome lattice, and considering a holomorphic map to C d with Zariski dense image). As Y is notcontained in D ∪ ∆ E , we can write h = ( h , h , h , h ) with h j : C → E holomorphic for each j ,satisfying:(i) Not all the h j are constant.(ii) ( h , h ) = ( h , h ) as holomorphic maps C → E , since Y is not included in ∆ E .(iii) The compositions P ◦ h j are well-defined elements of M , since Y is not contained in D .(iv) The following equation holds in M :( P ◦ h ) n + γ · ( P ◦ h ) n = ( P ◦ h ) n + γ · ( P ◦ h ) n since Y ⊆ Z and Z = ( f × f ) ∗ ∆ A k .By (i) and symmetry of the conditions (possibly replacing γ by 1 /γ ), we may assume that h isnon-constant. Hence P ◦ h is non-constant.We claim that P ◦ h is not the zero constant. For otherwise, (iv) would give(3.1) ( P ◦ h ) n + γ · ( P ◦ h ) n − γ · ( P ◦ h ) n = 0so that the holomorphic map [ P ◦ h : P ◦ h : P ◦ h ] : C → P would have image containedin a Fermat curve of degree n ≥
9. Such a curve has genus at least 28, so our holomorphic mapis constant by Picard’s theorem. This means that there are complex numbers c , c ∈ C with P ◦ h = c · P ◦ h and P ◦ h = c · P ◦ h . Furthermore, not both c and c are equal to 0, becauseof (3.1) and the fact that h is non-constant. We consider the two cases: • If c = 0 then h is non-constant. We get h = h and c = 1 because P is a stronguniqueness function. From (3.1) we get (1 + γ )( P ◦ h ) n = γ · ( P ◦ h ) n where γ = 0 and1 + γ = 0. Since P is a strong uniqueness function and h is non-constant, we deduce h = h . This gives 1 + γ = γ , impossible. • If c = 0 then h is non-constant. We get h = h and c = 1 because P is a stronguniqueness function. From (3.1) we get (1 − γ )( P ◦ h ) n = − γ · ( P ◦ h ) n and similarly wededuce h = h . This gives 1 − γ = − γ , impossible.This contradiction proves that P ◦ h is not the zero constant.We now claim that there is no c ∈ C × for which P ◦ h = c · P ◦ h . For the sake of contradiction,suppose there is such a c . Then h is non-constant because h is non-constant, and since P isa strong uniqueness function we deduce h = h and c = 1. Therefore P ◦ h = P ◦ h . Byitem (iv) and the condition γ = 0 we deduce ( P ◦ h ) n = ( P ◦ h ) n . If h or h is non-constant,so is the other and we deduce that they are equal because P is a strong uniqueness function;this would contradict item (ii) because we already know h = h . Therefore both h and h areconstant. If h , h are the same constant function, then we again get a contradiction with (ii),so they are different constant functions given by two different points q , q ∈ E respectively. Weobserve that q , q ∈ U ( k ) because Y ( k ) is Zariski dense in Y , which is not contained in D . Since P is injective on U ( k ), our hypothesis on n (the only n -th root of unity in k is 1) shows that( P ◦ h ) n = P ( q ) n = P ( q ) n = ( P ◦ h ) n , and we obtain a contradiction. This proves that there isno c ∈ C × for which P ◦ h = c · P ◦ h .Let us define the holomorphic map H : C → P by H = [ P ◦ h : P ◦ h : P ◦ h : P ◦ h ] nd observe that it is non-constant because P ◦ h is non-constant while P ◦ h is not of the form c · P ◦ h for any c ∈ C (the cases c = 0 and c = 0 are covered by the two previous claims).Furthermore, we observe that the image of H is contained in the Fermat surface F ⊆ P defined by(3.2) x n + γ · x n − x n − γ · x n = 0 . We recall that γ = 0 and n ≥
9. By Hayman’s theorem [12] the image of H must be contained inone of the “obvious” lines of F determined by a vanishing 2-terms sub-sum of (3.2). Since P ◦ h is not of the form c · P ◦ h for any c ∈ C , this only leaves the following possibilities: • P ◦ h = λ · P ◦ h with λ n = − γ . This gives that h is non-constant, and since P is a stronguniqueness function we would get h = h and λ = 1. This is not possible since γ = − • P ◦ h = λ · P ◦ h with λ n = γ . This gives that h is non-constant, and since P is a stronguniqueness function we would get h = h and λ = 1. This is not possible since γ = 1.This is a contradiction. Therefore Y must be contained in D ∪ ∆ E . (cid:3) Applications
We can now use the results of Sections 2 and 3 to prove the results stated in the introduction.
Proof of Theorem 1.1.
Let E be the projective closure of C , then E is an elliptic curve over k . Notethat P ∈ k ( E ) is a strong uniqueness function defined over k , by Theorem 2.1. Furthermore, P isregular on C ⊆ E , and we see that item (i) of Theorem 1.1 follows from Proposition 3.1.Item (ii) of Theorem 1.1 follows from Theorem 3.2. (cid:3) Proof of Corollary 1.2.
We choose a short Weierstrass equation for E and then we apply Theorem1.1 to the corresponding affine curve C ⊆ E . By item (i) of Theorem 1.1 we can shrink C to getan open set U as required by item (ii). Deleting finitely many k -rational points from U we get thedesired open set V . (cid:3) Proof of Corollary 1.3.
The elliptic curve E of affine equation y = x + x − E ( Q ) is isomorphic to Z , generated by the point (1 , E ( R ) is connected, hence E ( Q ) is dense in E ( R ) (its closure in the one-dimensional realLie group E ( R ) is open).Consider the morphism P : C → A Q given by P ( x, y ) = x + y . We claim that P induces aninjective map on real points C ( R ) → R . If not, we would have two different points q , q ∈ C ( R )such that the line through them has slope −
1. Hence, for some q ∈ C ( R ) (between q and q ) theline tangent to C ( R ) at q has slope −
1. However, it is a simple computation to check that the slopeof any non-vertical tangent of C ( R ) has absolute value larger than 2 . C ( R ) can have slope −
1, which proves injectivity on real points.Since P induces an injective function C ( R ) → R , it also induces an injective function C ( Q ) → Q .Thus, we can apply item (ii) of Theorem 1.1 with U = C . (cid:3) Acknowledgments
This research was supported by FONDECYT Regular grant 1190442.The results in this work were motivated by discussions at the PUC number theory seminar andI deeply thank the attendants. I also thank Gunther Cornelissen and Michael Zieve for commentson an earlier version of this work. eferences [1] T. An, J. Wang, P.-M. Wong, Strong uniqueness polynomials: the complex case . Complex Var. Theory Appl. 49(2004), no. 1, 25-54.[2] R. Avanzi, U. Zannier,
The equation f ( X ) = f ( Y ) in rational functions X = X ( t ) , Y = Y ( t ). Compositio Math.139 (2003), no. 3, 263-295.[3] D. Boneh, H. Corrigan-Gibbs. Bivariate polynomials modulo composites and their applications . Advances incryptology-ASIACRYPT 2014. Part I, 42-62, Lecture Notes in Comput. Sci., 8873, Springer, Heidelberg, 2014.[4] J. Browkin, J. Brzezi´nski,
Some remarks on the abc -conjecture . Math. Comput. 62, No.206, 931-939 (1994).[5] G. Cantor,
Beitr¨age zur Begr¨undung der transfiniten Mengenlehre . Math. Annalen 46 (1895), 481-512.[6] G. Cornelissen,
Stockage diophantien et hypoth`ese abc g´en´eralis´ee . C. R. Acad. Sci. Paris S´er. I Math. 328, no. 1(1999), 3-8.[7] G. Faltings,
Endlichkeitss¨atze f¨ur abelsche Variet¨aten ¨uber Zahlk¨orpern . Inventiones mathematicae 73.3 (1983):349-366.[8] G. Faltings,
Diophantine approximation on abelian varieties . Annals of Mathematics 133.3 (1991) 549-576.[9] G. Faltings,
The general case of S. Lang’s conjecture . In Barsotti Symposium in Algebraic Geometry, Perspec.Math, vol. 15, pp. 175-182. 1994.[10] R. Fueter, G. P´olya,
Rationale Abz¨ahlung der Gitterpunkte . Vierteljahrsschrift der Naturf. Gesellschaft in Z¨urich,58 (1923), 380-386.[11] H. Fujimoto,
On uniqueness of meromorphic functions sharing finite sets . Amer. J. Math. 122 (2000), no. 6,1175-1203.[12] W. Hayman,
Waring’s Problem f¨ur analytische Funktionen . Bayer. Akad. Wiss. Math. Natur. Kl. Sitzungsber.1984 (1985) 1-13.[13] P.-C. Hu, P. Li, C.-C. Yang,
Unicity of meromorphic mappings . Advances in Complex Analysis and its Applica-tions, 1. Kluwer Academic Publishers, Dordrecht, 2003. x+467 pp. ISBN: 1-4020-1219-5[14] J. Lew, A. Rosenberg,
Polynomial indexing of integer lattice-points. I. General concepts and quadratic polyno-mials . J. Number Theory 10 (1978), no. 2, 192-214.[15] J. Lew, A. Rosenberg,
Polynomial indexing of integer lattice-points. II. Nonexistence results for higher-degreepolynomials . J. Number Theory 10 (1978), no. 2, 215-243.[16] B. Poonen,
Multivariable polynomial injections on rational numbers . Acta Arithmetica, 2, no. 145 (2010): 123-127.[17] L.-P. Wang, X. Zhang,
Partial Bits Exposure Attacks on a New Commitment Scheme Based on the ZagierPolynomial . In International Conference on Information Security and Cryptology, pp. 357-366. Springer, Cham,2016.
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